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ruler-unit="cm"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="false" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="1876" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="1735" left="30" top="11.25" width="408" height="15.75" align-x="30" align-y="24" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="14"> 2.3. Метод, использующий разложение решения в ряд Тейлора. </f> </text> </region> <region region-id="1736" left="30" top="36.75" width="459" height="148.5" align-x="30" align-y="48" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr underline="false">Его можно также назвать методом припасовывания для решения краевых задач ОДУ.<sp count="2"/>Отличие этого метода от<sp count="2"/>предыдущих состоит в том, что исходное дифференциальное уравнение не надо<sp count="2"/>представлять в разностном виде. Искомое решение разбивается на ряд участков (число их в пределе должно стремиться к бесконечности), на каждом из которых оно представлено рядом Тейлора. Далее эти участки припасовываются (сшиваются). Алгоритм припасовывания описывается системой линейных алгебраических уравнений, имеющей, как правило, сильно разреженную матрицу. Именно эта матрица определяет пересчет краевых условий для каждого из участков. Решение же самой системы линейных алгебраических уравнений можно проводить методом Гаусса или другими методами, являющимися частными случаями последнего.</inlineAttr> </f> </text> </region> <region region-id="1406" left="24" top="192.75" width="447.75" height="54" align-x="24" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">По-прежнему будем рассматривать<sp count="2"/>систему линейных алгебраических уравнений</f> <f family="Arial Cyr" charset="204" size="12">с сильно разреженной матрицей, </f> <f family="Arial Cyr" charset="204" size="12">задающую алгоритм пересчета краевых условий</f> <f family="Arial Cyr" charset="204" size="12">от предыдущего участка к последующему<sp count="2"/>в общем виде и представляемую </f> <f family="Arial Cyr" charset="204" size="12">формулой (1)</f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="1413" left="36" top="258.75" width="276.75" height="17.25" align-x="75" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id 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xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">C</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="2"/> </region> <region region-id="1756" left="336" top="258.75" width="61.5" height="15.75" align-x="364.5" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:range xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">i</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">m</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> </math> <rendering item-idref="3"/> </region> <region region-id="1757" left="414" top="258.75" width="66" height="15.75" align-x="446.25" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:range xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">k</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">K</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> </math> <rendering item-idref="4"/> </region> <region region-id="1758" left="492" top="258.75" width="15.75" height="15.75" align-x="498.75" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </ml:parens> </math> <rendering item-idref="5"/> </region> <region region-id="1486" left="24" top="282.75" width="491.25" height="196.5" align-x="24" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Где m - порядок диф. ур.; К - число участков припасовывания или дискретных значений переменных, вводимое для численного решения задачи; i - порядок производной или номер краевого условия; </f> <f family="Arial Cyr" charset="204" size="12">k - номер участка припасовывания. </f> <region region-id="1417" left="467.25" top="311.25" width="21" height="17.25" align-x="477" align-y="322.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </math> <rendering item-idref="6"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> - дискретные значения искомого решения, записываемые в матрицу "Z". Смысл матриц А, В, ..., С будет пояснен ниже. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">В частном случае, когда заданы все<sp count="2"/>Z</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> i,0 </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> - начальные условия задачи Коши, формула (1) определяет их пересчет (алгоритм пересчета м.б. разным) в конечные значения Z(x) на участке припасовывания (или начальные значения последующего участка припасовывания). Вместо начальных условий могут быть заданы краевые условия. Отличие от начальных условий состоит в том, что какая-то часть из них задана вначале, а другая - в конце интервала изменения переменных. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> Ниже приводится алгоритм решения для случая 4-х диагональной матрицы. Надо сказать, что методы "стрельбы", "прогонки" являются частными случаями метода Гаусса. Параметры дифференциального уравнения следующие: m=2,<sp count="2"/>далее см. по тексту. </inlineAttr> </f> </text> </region> <region region-id="1428" left="18" top="486.75" width="500.25" height="93" align-x="18" align-y="498" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Описанный ниже алгоритм справедлив для дифференциального<sp count="2"/>уравнения (линей-</f> <f family="Arial Cyr" charset="204" size="12">ного): </f> <region region-id="1427" left="51.75" top="501" width="87" height="15.75" align-x="111.75" align-y="512.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">z''</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">P</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="7"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>с заданными граничными условиями:<sp count="2"/> </inlineAttr> </f> <region region-id="1429" left="353.25" top="501" width="71.25" height="17.25" align-x="392.25" align-y="512.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">Z</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> </math> <rendering item-idref="8"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>,<sp count="3"/>Z(x</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> K </f> <f family="Arial Cyr" charset="204" size="12">)=Z</f> <f family="Arial Cyr" charset="204" size="12">0,K </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12">или<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">Z'(x</f> <f family="Arial Cyr" charset="204" size="12"> K </f> <f family="Arial Cyr" charset="204" size="12">)=Z</f> <f family="Arial Cyr" charset="204" size="12"> 1 </f> <f family="Arial Cyr" charset="204" size="12"> ,K </f> <f family="Arial Cyr" charset="204" size="12">), но просто обобщается на случай уравнений более высокого порядка в том числе нелинейных, имеющих гладкое решение и разрешенных относительно стар-</f> <f family="Arial Cyr" charset="204" size="12">шей производной. </f> <f family="Arial Cyr" charset="204" size="12">Для участка с номером "k" справедливы следующие соотношения:</f> </text> </region> <region region-id="1431" left="24" top="591" width="429" height="35.25" align-x="69" align-y="612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:parens> <ml:id xml:space="preserve">x</ml:id> </ml:parens> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:sequence> <ml:real>2</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:sequence> <ml:real>3</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:parens> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:sequence> <ml:real>4</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>4</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:parens> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve" font="1">...</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="9"/> </region> <region region-id="1455" left="492" top="600.75" width="15.75" height="15.75" align-x="498.75" align-y="612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </ml:parens> </math> <rendering item-idref="10"/> </region> <region region-id="1784" left="90" top="636" width="179.25" height="33.75" align-x="90" align-y="636" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="11"/> <rendering item-idref="12"/> </region> <region region-id="1783" left="30" top="648.75" width="21" height="13.5" align-x="30" align-y="660" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Или</f> </text> </region> <region region-id="1787" left="24" top="708.75" width="482.25" height="33" align-x="24" align-y="720" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Здесь в (2) введены следующие обозначения: Z</f> <f family="Arial Cyr" charset="204" size="12"> 0,k </f> <f family="Arial Cyr" charset="204" size="12">(x) - функция от х;<sp count="2"/>z</f> <f family="Arial Cyr" charset="204" size="12"> i,k </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">/</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">i! - коэффициент</f> <f family="Arial Cyr" charset="204" size="12">ряда Тейлора функции </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> 0,k </f> <f family="Arial Cyr" charset="204" size="12">(x). Отсюда из уравнения (2) формально можно записать</f> </text> </region> <region region-id="1792" left="36" top="762" width="204.75" height="33" align-x="36" align-y="762" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="13"/> <rendering item-idref="14"/> </region> <region region-id="1793" left="30" top="810.75" width="393.75" height="16.5" align-x="30" align-y="822" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">и получить значение первой производной от </f> <f family="Arial Cyr" charset="204" size="12">функции </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> 0,k </f> <f family="Arial Cyr" charset="204" size="12">(x) на участке k</f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> </text> </region> <region region-id="1434" left="24" top="843" width="349.5" height="35.25" align-x="70.5" align-y="864" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:parens> <ml:id xml:space="preserve">x</ml:id> </ml:parens> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:sequence> <ml:real>2</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:sequence> <ml:real>3</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:sequence> <ml:real>4</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:parens> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve" font="1">...</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="15"/> </region> <region region-id="1454" left="492" top="852.75" width="15.75" height="15.75" align-x="498.75" align-y="864" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>3</ml:real> </ml:parens> </math> <rendering item-idref="16"/> </region> <region region-id="1794" left="30" top="882" width="477" height="59.25" align-x="30" align-y="894" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Именно эти выражения (2)-(3) необходимо пересчитывать на каждом участке </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12"> из </f> <f family="Arial Cyr" charset="204" size="12">начальных<sp count="2"/>Z</f> <f family="Arial Cyr" charset="204" size="12"> i,k </f> <f family="Arial Cyr" charset="204" size="12">в конечные</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> i,k+1 </f> <f family="Arial Cyr" charset="204" size="12">, где<sp count="2"/>i=0,1,...(n-1),<sp count="3"/> </f> <f family="Arial Cyr" charset="204" size="12">n</f> <f family="Arial Cyr" charset="204" size="12"> - порядок дифференциального</f> <f family="Arial Cyr" charset="204" size="12">уравнения или число задаваемых краевых условий. Коротко первую производную</f> <f family="Arial Cyr" charset="204" size="12">на участке<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12"> можно записать так:</f> </text> </region> <region region-id="1795" left="156" top="956.25" width="116.25" height="37.5" align-x="156" align-y="978" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <region region-id="1724" left="156" top="957" width="119.25" height="36.75" align-x="202.5" align-y="978" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:parens> <ml:id xml:space="preserve">x</ml:id> </ml:parens> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:parens> <ml:id xml:space="preserve">x</ml:id> </ml:parens> </ml:apply> </ml:parens> </ml:lambda> </ml:apply> </ml:apply> </math> <rendering item-idref="17"/> </region> </text> </region> <region region-id="1796" left="36" top="996.75" width="474" height="46.5" align-x="36" align-y="1008" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Значения коэффициентов ряда Тейлора функции </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> 0,k </f> <f family="Arial Cyr" charset="204" size="12">(x) (2) определяются из началь- ных (краевых) условий и путем дифференцирования исходного уравнения с исклю- чением старшей производной и подстановкой x=x</f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12">как показано ниже</f> </text> </region> <region region-id="1438" left="90" top="1050.75" width="102" height="17.25" align-x="121.5" align-y="1062" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>2</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="18"/> </region> <region region-id="1439" left="216" top="1050.75" width="159" height="17.25" align-x="247.5" align-y="1062" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>3</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f'</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P'</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="19"/> </region> <region region-id="1453" left="492" top="1056.75" width="15.75" height="15.75" align-x="498.75" align-y="1068" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>4</ml:real> </ml:parens> </math> <rendering item-idref="20"/> </region> <region region-id="1442" left="96" top="1086.75" width="275.25" height="17.25" align-x="127.5" align-y="1098" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>4</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f''</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P''</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P'</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="21"/> </region> <region region-id="1458" left="492" top="1086.75" width="15.75" height="15.75" align-x="498.75" align-y="1098" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>5</ml:real> </ml:parens> </math> <rendering item-idref="22"/> </region> <region region-id="1800" left="108" top="1122" width="125.25" height="33" align-x="108" align-y="1122" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="23"/> <rendering item-idref="24"/> </region> <region region-id="1801" left="30" top="1134.75" width="45" height="13.5" align-x="30" align-y="1146" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Причем:</f> </text> </region> <region region-id="1464" left="18" top="1170" width="485.25" height="36" align-x="18" align-y="1182" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Учитывая. что<sp count="2"/> </f> <region region-id="1462" left="104.25" top="1170.75" width="69" height="17.25" align-x="134.25" align-y="1182" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="25"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> , Z</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> i,k </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">(x</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> k+1 </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">)=Z</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">i,k+1</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="3"/>и, подставляя в (2), (3) выражения</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> (4), (5), получим (для k=0..K-1):</inlineAttr> </f> </text> </region> <region region-id="1465" left="30" top="1218.75" width="174" height="17.25" align-x="72.75" align-y="1230" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">B</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">C</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="26"/> </region> <region region-id="1495" left="222" top="1218.75" width="174" height="17.25" align-x="264.75" align-y="1230" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">D</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">E</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">G</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="27"/> </region> <region region-id="1496" left="408" top="1218.75" width="66" height="15.75" align-x="440.25" align-y="1230" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:range xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">k</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">K</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> </math> <rendering item-idref="28"/> </region> <region region-id="1470" left="492" top="1218.75" width="15.75" height="15.75" align-x="498.75" align-y="1230" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>6</ml:real> </ml:parens> </math> <rendering item-idref="29"/> </region> <region region-id="1725" left="30" top="1240.5" width="420.75" height="79.5" align-x="30" align-y="1266" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Где:<sp count="2"/> </f> <region region-id="1726" left="60" top="1241.25" width="229.5" height="39" align-x="81.75" align-y="1266" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P'</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P''</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="30"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>,<sp count="2"/> </inlineAttr> </f> <region region-id="1728" left="306.75" top="1241.25" width="130.5" height="39" align-x="327" align-y="1266" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">B</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P'</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="31"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> , </inlineAttr> </f> <region region-id="1729" left="30" top="1281" width="194.25" height="39" align-x="51" align-y="1305.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">C</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f'</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f''</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="32"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ,<sp count="2"/> </inlineAttr> </f> <region region-id="1731" left="243" top="1284.75" width="57" height="36.75" align-x="264" align-y="1305.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">D</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:lambda> </ml:apply> </ml:apply> </math> <rendering item-idref="33"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> , </f> </inlineAttr> <region region-id="1732" left="312.75" top="1284.75" width="54.75" height="36.75" align-x="332.25" align-y="1305.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">E</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">B</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:lambda> </ml:apply> </ml:apply> </math> <rendering item-idref="34"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> ,<sp count="2"/> </f> </inlineAttr> <region region-id="1733" left="384" top="1284.75" width="57" height="36.75" align-x="405" align-y="1305.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">G</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">C</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:lambda> </ml:apply> </ml:apply> </math> <rendering item-idref="35"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/> </inlineAttr> </f> </text> </region> <region region-id="1473" left="24" top="1326.75" width="491.25" height="54" align-x="24" align-y="1338" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Здесь коэффициенты A, B, C, D, E, G известны и будут определяться таким образом ниже при решении конкретных задач. Формулы (6) далее будут использованы как рекуррентные при нахождении решения краевой задачи. Они представляют собой систему линейных алгебраических уравнений в виде 4-х диагональной матрицы. </f> </text> </region> <region region-id="1867" left="30" top="1410.75" width="420" height="27" align-x="30" align-y="1422" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Решать (6) можно методом Гаусса, последовательно исключая неизвестные переменные.</f> </text> </region> <region region-id="1859" left="30" top="1452.75" width="461.25" height="63" align-x="30" align-y="1464" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">В первой паре уравнений (6)<sp count="2"/>известным будет Z</f> <f family="Arial Cyr" charset="204" size="12"> 0,0 </f> <f family="Arial Cyr" charset="204" size="12"> . Следовательно,<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">значения</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> 0,1<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">и</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> 1 </f> <f family="Arial Cyr" charset="204" size="12"> ,1 </f> <f family="Arial Cyr" charset="204" size="12">можно будет выразить через </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> 1 </f> <f family="Arial Cyr" charset="204" size="12"> ,0 </f> <f family="Arial Cyr" charset="204" size="12">. Этот процесс исключения </f> <f family="Arial Cyr" charset="204" size="12">можно продолжить для второй и последующих пар и всякий раз выражать </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12"> ,k<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">через</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> 1 </f> <f family="Arial Cyr" charset="204" size="12"> ,0 </f> <f family="Arial Cyr" charset="204" size="12">. Предположим это нам удалось сделать для<sp count="2"/>(k-1)-й пары<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">уравнений: </f> </text> </region> <region region-id="1474" left="54" top="1530" width="156.75" height="18" align-x="94.5" align-y="1542" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="36"/> </region> <region region-id="1476" left="246" top="1530.75" width="41.25" height="15.75" align-x="275.25" align-y="1542" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:range xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">i</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:range> </math> <rendering item-idref="37"/> </region> <region region-id="1860" left="480" top="1530.75" width="15.75" height="15.75" align-x="486.75" align-y="1542" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>7</ml:real> </ml:parens> </math> <rendering item-idref="38"/> </region> <region region-id="1481" left="24" top="1553.25" width="447" height="127.5" align-x="24" align-y="1566" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Тогда, подставляя (7) в (6), получим: </f> <region region-id="1482" left="234" top="1554" width="123" height="18" align-x="263.25" align-y="1566" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="39"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> . Сравнивая результат подстановки с последним выражением, получим рекуррентные формулы вычисления коэффициентов </inlineAttr> </f> <region region-id="1483" left="244.5" top="1586.25" width="24" height="16.5" align-x="255.75" align-y="1598.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">α</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:sequence> </math> <rendering item-idref="40"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> (всё это справедливо для линейных </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">уравнений)</inlineAttr> </f> <region region-id="1572" left="24" top="1630.5" width="185.25" height="18" align-x="57" align-y="1642.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="41"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ,<sp count="3"/> </inlineAttr> </f> <region region-id="1573" left="225" top="1630.5" width="221.25" height="18" align-x="257.25" align-y="1642.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="42"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ,</inlineAttr> </f> <region region-id="1574" left="24" top="1649.25" width="184.5" height="18" align-x="57" align-y="1661.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">E</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="43"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/>,<sp count="3"/> </f> </inlineAttr> <region region-id="1575" left="228" top="1649.25" width="220.5" height="18" align-x="260.25" align-y="1661.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">E</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">G</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="44"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">Ниже будет приведен алгоритм вычисления этих </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12">коэффициентов.</f> </text> </region> <region region-id="1802" left="492" top="1626.75" width="21" height="15.75" align-x="501.75" align-y="1638" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:mult style="auto-select"/> <ml:real>7</ml:real> <ml:id xml:space="preserve">a</ml:id> </ml:apply> </ml:parens> </math> <rendering item-idref="45"/> </region> <region region-id="1354" left="24" top="1680.75" width="487.5" height="73.5" align-x="24" align-y="1698" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Приведем</f> <f family="Arial Cyr" charset="204" size="12"> пример </f> <f family="Arial Cyr" charset="204" size="12"> решения уравнения </f> <region region-id="49" left="247.5" top="1681.5" width="60.75" height="21" align-x="282.75" align-y="1698" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">z''</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="46"/> </region> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/>с граничными условиями: </f> <region region-id="54" left="462.75" top="1686.75" width="42.75" height="15.75" align-x="490.5" align-y="1698" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="47"/> </region> <f family="Arial Cyr" charset="204" size="12"> ,<sp count="2"/> </f> <region region-id="55" left="33.75" top="1703.25" width="52.5" height="34.5" align-x="71.25" align-y="1723.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> </math> <rendering item-idref="48"/> </region> <f family="Arial Cyr" charset="204" size="12"> <sp count="3"/>(другим примером может быть z'' + p(t)*z = f(t), где p(t)=t^2 или t, </f> <f family="Arial Cyr" charset="204" size="12">f(t)=sin(t)*e</f> <f family="Arial Cyr" charset="204" size="12">-t</f> <f family="Arial Cyr" charset="204" size="12">).</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="3"/>Точное решение этого уравнения есть:</f> </text> </region> <region region-id="1577" left="24" top="1779" width="230.25" height="47.25" align-x="53.25" align-y="1812" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">yt</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="49"/> </region> <region region-id="1580" left="336" top="1800.75" width="53.25" height="15.75" align-x="367.5" align-y="1812" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:provenance expr-id="1"> <originRef doc-id="66f65496-debc-41ba-835b-b3474937ab71" version-id="1c6b8a94-b7d0-4499-84b2-8f948851f9b2" branch-id="00000000-0000-0000-0000-000000000000" revision-num="25" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЧислМетоды\КраевыеЗадачи\МетСтрельбы.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="66f65496-debc-41ba-835b-b3474937ab71" version-id="1c6b8a94-b7d0-4499-84b2-8f948851f9b2" branch-id="00000000-0000-0000-0000-000000000000" revision-num="25" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЧислМетоды\КраевыеЗадачи\МетСтрельбы.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">72f877ffa76c567a47fae3ce36224632</contentHash> <ml:apply> <ml:id xml:space="preserve">yt</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:provenance> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="50"> <element-image-map> <box left="1.5" top="0.75" width="23.25" height="14.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="1581" left="414" top="1791.75" width="63" height="34.5" align-x="455.25" align-y="1812" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">yt</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </result> </ml:eval> </math> <rendering item-idref="51"/> </region> <region region-id="1872" left="30" top="1848" width="89.25" height="14.25" align-x="30" align-y="1860" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Пусть заданы: </f> </text> </region> <region region-id="1868" left="144" top="1848.75" width="35.25" height="17.25" align-x="162.75" align-y="1860" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define 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<ml:real>0</ml:real> <ml:real>2</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="58"/> </region> <region region-id="1424" left="36" top="1923" width="56.25" height="35.25" align-x="49.5" align-y="1944" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">b</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">KK</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="59"/> </region> <region region-id="1423" left="114" top="1927.5" width="99" height="21" align-x="126.75" align-y="1944" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math 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<math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">a</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="64"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <region region-id="1502" left="193.5" top="1974.75" width="48.75" height="17.25" align-x="226.5" align-y="1986" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:id xml:space="preserve">a</ml:id> </ml:define> </math> <rendering item-idref="65"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ,<sp count="2"/> </inlineAttr> </f> <region region-id="1503" left="257.25" top="1974.75" width="57.75" height="17.25" align-x="298.5" align-y="1986" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <ml:real>2</ml:real> </ml:define> </math> <rendering item-idref="66"/> </region> </text> </region> <region region-id="1875" left="30" top="2004.75" width="368.25" height="13.5" align-x="30" align-y="2016" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">3)<sp count="2"/>дискретные значения функций, вычисленные в 3*(КК+1) - точках.</f> </text> </region> <region region-id="1295" left="42" top="2028.75" width="39" height="17.25" align-x="48" align-y="2040" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">p</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> </math> <rendering item-idref="67"/> </region> <region region-id="1297" left="96" top="2028.75" width="36.75" height="17.25" align-x="99.75" align-y="2040" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> </math> <rendering item-idref="68"/> </region> <region region-id="1300" left="30" top="2058.75" width="462" height="13.5" align-x="30" align-y="2070" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">4)<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">дискретные значения коэффициентов, вычисленные в</f> <f family="Arial Cyr" charset="204" size="12"> (КК+1) - точках:</f> </text> </region> <region region-id="1275" left="24" top="2117.25" 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xml:space="preserve">h</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">AA</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:define> </math> <rendering item-idref="70"/> </region> <region region-id="1278" left="18" top="2183.25" width="361.5" height="39" align-x="69.75" align-y="2208" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">CC</ml:id> <ml:boundVars> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:real>2</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">p</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="71"/> </region> <region region-id="1279" left="396" top="2187" width="120" height="36.75" align-x="448.5" align-y="2208" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">GG</ml:id> <ml:boundVars> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">CC</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:define> </math> <rendering item-idref="72"/> </region> <region region-id="1280" left="18" top="2243.25" width="207.75" height="39" align-x="60.75" align-y="2268" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">B</ml:id> <ml:boundVars> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">p</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:id xml:space="preserve">p</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="73"/> </region> <region region-id="1281" left="258" top="2247" width="100.5" height="36.75" align-x="300" align-y="2268" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">E</ml:id> <ml:boundVars> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">B</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:define> </math> <rendering item-idref="74"/> </region> <region region-id="1366" left="18" top="2297.25" width="468.75" height="51" align-x="18" align-y="2310" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Зададим также для единообразия: </f> <region region-id="1303" left="216.75" top="2298" width="50.25" height="18" align-x="250.5" align-y="2310" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="75"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>, </inlineAttr> </f> <region region-id="1304" left="280.5" top="2298" width="50.25" height="18" align-x="314.25" align-y="2310" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="76"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ,<sp count="2"/> </inlineAttr> </f> <region region-id="1306" left="344.25" top="2298" width="48.75" height="18" align-x="377.25" align-y="2310" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:id xml:space="preserve">a</ml:id> </ml:define> </math> <rendering item-idref="77"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> , </inlineAttr> </f> <region region-id="1307" left="402.75" top="2298" width="49.5" height="18" align-x="435.75" align-y="2310" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="78"/> </region> <f family="Arial Cyr" charset="204" size="12">Т</f> <f family="Arial Cyr" charset="204" size="12">аким образом, получили:</f> <f family="Arial Cyr" charset="204"> <sp/> </f> <region region-id="1511" left="169.5" top="2316.75" width="133.5" height="18" align-x="201.75" align-y="2328.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="79"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/>,<sp count="2"/> </f> </inlineAttr> <region region-id="1512" left="319.5" top="2316.75" width="133.5" height="18" align-x="351.75" align-y="2328.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="80"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> ,<sp count="3"/> </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12">и вычислим:</f> </text> </region> <region region-id="1365" left="36" top="2364" width="277.5" height="130.5" align-x="53.25" align-y="2376" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">α</ml:id> <ml:program> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:real>0</ml:real> </ml:localDefine> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:real>1</ml:real> </ml:localDefine> <ml:for> <ml:id xml:space="preserve">k</ml:id> <ml:range> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">KK</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> <ml:program> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">AA</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">B</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:localDefine> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">D</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:localDefine> </ml:program> </ml:for> <ml:id xml:space="preserve">α</ml:id> </ml:program> </ml:define> </math> <rendering item-idref="81"/> </region> <region region-id="1695" left="36" top="2520" width="331.5" height="130.5" align-x="52.5" align-y="2532" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">β</ml:id> <ml:program> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:id xml:space="preserve">a</ml:id> </ml:localDefine> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:real>0</ml:real> </ml:localDefine> <ml:for> <ml:id xml:space="preserve">k</ml:id> <ml:range> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">KK</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> <ml:program> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">AA</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">B</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">CC</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:localDefine> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">D</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">GG</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:localDefine> </ml:program> </ml:for> <ml:id xml:space="preserve">β</ml:id> </ml:program> </ml:define> </math> <rendering item-idref="82"/> </region> <region region-id="1589" left="36" top="2670.75" width="373.5" height="13.5" align-x="36" align-y="2682" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Для визуальной проверки выведем на дисплей некоторые значения:</f> </text> </region> <region region-id="1764" left="36" top="2700" width="89.25" height="18" align-x="76.5" align-y="2712" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.60393978861178022</ml:real> </result> </ml:eval> </math> <rendering item-idref="83"/> </region> <region region-id="1763" left="126" top="2700" width="96" height="18" align-x="166.5" align-y="2712" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.6039397877543341</ml:real> </result> </ml:eval> </math> <rendering item-idref="84"/> </region> <region region-id="1761" left="252" top="2700" width="63" height="18" align-x="293.25" align-y="2712" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.0000000000908043</ml:real> </result> </ml:eval> </math> <rendering item-idref="85"/> </region> <region region-id="1765" left="336" top="2695.5" width="126.75" height="22.5" align-x="377.25" align-y="2712" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2.6129656402994195E-10</ml:real> </result> </ml:eval> </math> <rendering item-idref="86"/> </region> <region region-id="1876" left="24" top="2729.25" width="490.5" height="65.25" align-x="24" align-y="2742" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Заметим, что коэффициенты </f> <region region-id="1712" left="191.25" top="2730" width="53.25" height="18" align-x="216" align-y="2742" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:sequence> </math> <rendering item-idref="87"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>, в данном примере не зависят от краевых (граничных) условий. Коэффициент </inlineAttr> </f> <region region-id="1713" left="228" top="2748.75" width="25.5" height="18" align-x="240" align-y="2760.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </math> <rendering item-idref="88"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> , зависит от Y</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> 0,0 </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> таким образом: при </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">Y</f> <f family="Arial Cyr" charset="204" size="12">0,0</f> <f family="Arial Cyr" charset="204" size="12">=0</f> <region region-id="1715" left="24" top="2772" width="82.5" height="18" align-x="64.5" align-y="2784" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <ml:real>0.3961</ml:real> </ml:apply> </math> <rendering item-idref="89"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">, при </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12">Y</f> <f family="Arial Cyr" charset="204" size="12">0,0</f> <f family="Arial Cyr" charset="204" size="12">=1 </f> <region region-id="1716" left="176.25" top="2772" width="101.25" height="18" align-x="216.75" align-y="2784" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:minus/> <ml:real>0.3961</ml:real> <ml:real>1</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="90"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ,<sp count="2"/>при </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">Y</f> <f family="Arial Cyr" charset="204" size="12">0,0</f> <f family="Arial Cyr" charset="204" size="12">=2<sp count="2"/> </f> <region region-id="1717" left="357" top="2772" width="101.25" height="18" align-x="397.5" align-y="2784" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:minus/> <ml:real>0.3961</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="91"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> и<sp count="2"/>т. д.</inlineAttr> </f> </text> </region> <region region-id="1770" left="36" top="2814.75" width="183.75" height="13.5" align-x="36" align-y="2826" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Напомним граничные условия:</f> </text> </region> <region region-id="1769" left="234" top="2814.75" width="75" height="17.25" align-x="260.25" align-y="2826" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">KK</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.5707963267948966</ml:real> </result> </ml:eval> </math> <rendering item-idref="92"/> </region> <region region-id="1768" left="318" top="2814.75" width="54" height="17.25" align-x="350.25" align-y="2826" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="93"/> </region> <region region-id="1767" left="384" top="2814.75" width="62.25" height="17.25" align-x="424.5" align-y="2826" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </result> </ml:eval> </math> <rendering item-idref="94"/> </region> <region region-id="1622" left="36" top="2886.75" width="295.5" height="16.5" align-x="36" align-y="2898" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Исходя из (7), если задана функция Y</f> <f family="Arial Cyr" charset="204" size="12"> 0,KK </f> <f family="Arial Cyr" charset="204" size="12">, </f> <f family="Arial Cyr" charset="204" size="12"> вычисляем:</f> </text> </region> <region region-id="1804" left="360" top="2876.25" width="125.25" height="38.25" align-x="393" align-y="2898" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="95"/> </region> <region region-id="1631" left="36" top="2929.5" width="437.25" height="48.75" align-x="36" align-y="2952" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Если же задана её производная Y</f> <f family="Arial Cyr" charset="204" size="12"> 1,KK </f> <f family="Arial Cyr" charset="204" size="12">, то вычисляем:<sp count="3"/> </f> <region region-id="1632" left="345.75" top="2930.25" width="126.75" height="38.25" align-x="378.75" align-y="2952" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="96"/> </region> </text> </region> <region region-id="1648" left="30" top="2994.75" width="94.5" height="13.5" align-x="30" align-y="3006" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Вычислим далее:</f> </text> </region> <region region-id="1649" left="138" top="2994.75" width="58.5" height="15.75" align-x="152.25" align-y="3006" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">k</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="97"/> </region> <region region-id="1650" left="210" top="2994" width="135.75" height="18" align-x="243" align-y="3006" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="98"/> </region> <region region-id="1651" left="360" top="2994" width="135.75" height="18" align-x="393" align-y="3006" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="99"/> </region> <region region-id="1647" left="30" top="3024.75" width="180.75" height="13.5" align-x="30" align-y="3036" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Выведем на дисплей значение:</f> </text> </region> <region region-id="1745" left="216" top="3024.75" width="96" height="17.25" align-x="256.5" align-y="3036" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.60393978738954834</ml:real> </result> </ml:eval> </math> <rendering item-idref="100"/> </region> <region region-id="1771" left="30" top="3054.75" width="474.75" height="27" align-x="30" align-y="3066" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Остальные компоненты представлены в виде разворачиваемой в электронной версии пособия матрицы Y.</f> </text> </region> <region region-id="1652" left="18" top="3125.25" width="465.75" height="39.75" align-x="33.75" align-y="3144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Y</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="2" cols="1001"> <ml:real>1</ml:real> <ml:real>1.3960602112614515</ml:real> <ml:real>1.0021929247043186</ml:real> <ml:real>1.3960572558873</ml:real> <ml:real>1.0043858401255794</ml:real> <ml:real>1.3960483923522911</ml:real> <ml:real>1.006578736986318</ml:real> <ml:real>1.3960336245449576</ml:real> <ml:real>1.0087716060151841</ml:real> 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<ml:real>2.0037592743154544</ml:real> <ml:real>-0.5926718274883308</ml:real> <ml:real>2.0028260938727591</ml:real> <ml:real>-0.59549025617531359</ml:real> <ml:real>2.0018884870008642</ml:real> <ml:real>-0.59830773089437406</ml:real> <ml:real>2.0009464552043532</ml:real> <ml:real>-0.60112424388480667</ml:real> <ml:real>2</ml:real> <ml:real>-0.60393978738954834</ml:real> </ml:matrix> </result> </ml:eval> <resultFormat> <table item-idref="101"/> </resultFormat> </math> <rendering item-idref="102"/> </region> <region region-id="1772" left="42" top="3192.75" width="405.75" height="13.5" align-x="42" align-y="3204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Полученные приближенное и точное решения можно сравнить на графике:</f> </text> </region> <region region-id="1773" left="42" top="3264" width="255" height="135.75" align-x="42" align-y="3264" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="103"/> <rendering item-idref="104"/> </region> <region region-id="1776" left="390" top="3312.75" width="77.25" height="17.25" align-x="418.5" align-y="3324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:real>700</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.0995574287564276</ml:real> </result> </ml:eval> </math> <rendering item-idref="105"/> </region> <region region-id="1775" left="384" top="3354.75" width="91.5" height="17.25" align-x="426.75" align-y="3366" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>700</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2.0829064863423685</ml:real> </result> </ml:eval> </math> <rendering item-idref="106"/> </region> <region region-id="1746" left="42" top="3456" width="465" height="27.75" align-x="42" align-y="3468" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12"> Упражнение. </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false"> Привести п</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false">ример </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false">других краевых условий, когда частного решения нет<sp count="8"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false">(</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false">М</inlineAttr> </f> <f family="Arial Cyr" charset="204">ожно взять граничные условия: y(0)=1, y'(Pi/2)=1).</f> </text> </region> <region region-id="1750" left="30" top="3510.75" width="463.5" height="27" align-x="30" align-y="3522" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Если Вы хотите сравнить полученное Вами решение с ответом, откройте в электрон-</f> <f family="Arial Cyr" charset="204" size="12">ной версии пособия ниже представленную закрытую область.</f> </text> </region> <region region-id="1753" left="0" top="3543" width="6008.25" align-x="8.25" align-y="3552" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="" height="1388.25"> <area is-collapsed="false" name="" show-name="false" show-border="true" show-icon="true" show-timestamp="true" allow-expand="false" is-locked="false" timestamp="" top-lock-id="1753" bottom-lock-id="1754" bottom-tag=""> <region region-id="1864" left="30" top="3561.75" width="470.25" height="16.5" align-x="30" align-y="3576" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">По-прежнему рассматриваем уравнение y'' + y = e</f> <f family="Arial Cyr" charset="204" size="12"> -x </f> <sp/> <f family="Arial Cyr" charset="204" size="12">и для заданных граничных условий</f> </text> </region> <region region-id="1806" left="24" top="3591.75" width="489.75" height="136.5" align-x="24" align-y="3612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0" size="10"> <inlineAttr font-style="normal" underline="false"> y(0)=1, y'(Pi/2)=1 </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false">его решение будет</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-style="normal" underline="false">: </inlineAttr> </f> <region region-id="1538" left="215.25" top="3592.5" width="179.25" height="33.75" align-x="244.5" align-y="3612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="107"/> </region> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">Где не</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">известный коэффициент В определяется из 2-го граничного условия:</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="10"> <sp/> </f> </inlineAttr> <region region-id="1539" left="24" top="3654" width="183.75" height="33.75" align-x="54" align-y="3673.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">y'</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:neg/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="108"/> </region> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">,<sp count="2"/> </inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">то есть</f> </inlineAttr> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <region region-id="1540" left="266.25" top="3640.5" width="145.5" height="47.25" align-x="306.75" align-y="3673.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">y'</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:lessThan/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>-1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:real>0</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="109"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="10">.</f> </inlineAttr> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">Это означает, что</f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, </inlineAttr> </f> <f family="Arial" charset="0">y'(Pi/2) <>1! </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false">Следовательно</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="10"> <inlineAttr font-style="normal" underline="false">, </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false">какой бы</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-style="normal" underline="false"> <sp/> </inlineAttr> </f> <inlineAttr font-style="normal" underline="false"> <f family="Arial Cyr" charset="204" size="12">коэффициент В мы не выбирали, полученное таким образом решение решение заданным краевым условиям удовлетворять не будет. Представим решение как функцию параметра В</f> </inlineAttr> </text> </region> <region region-id="1554" left="30" top="3790.5" width="192" height="33.75" align-x="69.75" align-y="3810" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">y</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">B</ml:id> </ml:boundVars> </ml:function> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:define> </math> <rendering item-idref="110"/> </region> <region region-id="1553" left="306" top="3777" width="137.25" height="47.25" align-x="387.75" align-y="3810" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>-1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.603939788175381</ml:real> </result> </ml:eval> </math> <rendering item-idref="111"/> </region> <region region-id="1808" left="30" top="3840.75" width="483" height="54" align-x="30" align-y="3852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">и построим графики y(t,B) для разных В. Все эти и другие кривые в точке t=pi/2, будут иметь одну и ту же касательную с угловым коэффициентом - 0,6039. Кроме этого у</f> <f family="Arial Cyr" charset="204" size="12">множества решений </f> <f family="Arial Cyr" charset="204" size="12">y(t,B) есть стационарная точка t=pi, через которую проходит каж-</f> <f family="Arial Cyr" charset="204" size="12">дая кривая этого семейства.</f> </text> </region> <region region-id="1556" left="36" top="3918" width="251.25" height="139.5" align-x="36" align-y="3918" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="112"/> <rendering item-idref="113"/> </region> <region region-id="1679" left="18" top="4092" width="490.5" height="31.5" align-x="18" align-y="4104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Сформулируем запись граничных условий</f> <f family="Arial" charset="0" size="12">: </f> <region region-id="1564" left="264" top="4092.75" width="49.5" height="17.25" align-x="297" align-y="4104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="114"/> </region> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ,<sp count="2"/> </inlineAttr> </f> <region region-id="1565" left="328.5" top="4092.75" width="57.75" height="17.25" align-x="369.75" align-y="4104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="115"/> </region> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">и найдем коэффициенты альфа и бетта для участка k=0</inlineAttr> </f> </text> </region> <region region-id="1826" left="24" top="4140" width="50.25" height="18" align-x="57.75" align-y="4152" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="116"/> </region> <region region-id="1827" left="102" top="4140" width="50.25" height="18" align-x="135.75" align-y="4152" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="117"/> </region> <region region-id="1828" left="180" top="4140" width="48.75" height="18" align-x="213" align-y="4152" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:id xml:space="preserve">a</ml:id> </ml:define> </math> <rendering item-idref="118"/> </region> <region region-id="1829" left="258" top="4140" width="49.5" height="18" align-x="291" align-y="4152" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="119"/> </region> <region region-id="1830" left="24" top="4164.75" width="422.25" height="13.5" align-x="24" align-y="4176" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">На участке k=0 решение и его производная будут определяться уравнениями</f> </text> </region> <region region-id="1831" left="18" top="4211.25" width="468.75" height="18.75" align-x="18" align-y="4224" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0" size="12"> <sp/> </f> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <region region-id="1570" left="24" top="4212" width="129.75" height="18" align-x="55.5" align-y="4224" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="120"/> </region> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>,<sp count="4"/> </inlineAttr> </f> <region region-id="1571" left="177.75" top="4212" width="130.5" height="18" align-x="208.5" align-y="4224" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="121"/> </region> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/> </inlineAttr> </f> <f family="Arial" charset="0" size="12"> <sp/> </f> </text> </region> <region region-id="1832" left="24" top="4242.75" width="468" height="46.5" align-x="24" align-y="4254" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Так как Z</f> <f family="Arial Cyr" charset="204" size="12"> 1,0 </f> <f family="Arial Cyr" charset="204" size="12"> неизвестно на начальном этапе и задается только как граничное условие </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> 1,k </f> <f family="Arial Cyr" charset="204" size="12"> в конце траектории описывающей решение, то, используя уранение (7), для </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> 1,0 </f> <f family="Arial Cyr" charset="204" size="12">получим</f> </text> </region> <region region-id="1654" left="138" top="4316.25" width="125.25" height="38.25" align-x="171" align-y="4338" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="122"/> </region> <region region-id="1665" left="312" top="4321.5" width="110.25" height="22.5" align-x="344.25" align-y="4338" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>6138388362.3151617</ml:real> </result> </ml:eval> </math> <rendering item-idref="123"/> </region> <region region-id="1833" left="36" top="4368.75" width="434.25" height="27" align-x="36" align-y="4380" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Проведем визуализацию значений коэффициентов альфа и бетта, которые они</f> <f family="Arial Cyr" charset="204" size="12">принимают в конце интервала (траектории)</f> </text> </region> <region region-id="1841" left="42" top="4416" width="63" height="18" align-x="83.25" align-y="4428" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.0000000000908043</ml:real> </result> </ml:eval> </math> <rendering item-idref="124"/> </region> <region region-id="1840" left="120" top="4411.5" width="126.75" height="22.5" align-x="161.25" align-y="4428" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2.6129656402994195E-10</ml:real> </result> </ml:eval> </math> <rendering item-idref="125"/> </region> <region region-id="1839" left="264" top="4416" width="89.25" height="18" align-x="304.5" align-y="4428" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.60393978861178022</ml:real> </result> </ml:eval> </math> <rendering item-idref="126"/> </region> <region region-id="1838" left="366" top="4416" width="96" height="18" align-x="406.5" align-y="4428" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.6039397877543341</ml:real> </result> </ml:eval> </math> <rendering item-idref="127"/> </region> <region region-id="1847" left="30" top="4452.75" width="407.25" height="30" align-x="30" align-y="4464" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Также вычислим вектор решения {Y</f> <f family="Arial Cyr" charset="204" size="12"> 0,k </f> <f family="Arial Cyr" charset="204" size="12">}<sp count="2"/>и вектор его производной<sp count="2"/>{Y</f> <f family="Arial Cyr" charset="204" size="12"> 1,k </f> <f family="Arial Cyr" charset="204" size="12">} по </f> <f family="Arial Cyr" charset="204" size="12">уравнению (7)</f> </text> </region> <region region-id="1844" left="42" top="4500.75" width="58.5" height="15.75" align-x="56.25" align-y="4512" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">k</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="128"/> </region> <region region-id="1845" left="132" top="4500" width="135.75" height="18" align-x="165" align-y="4512" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="129"/> </region> <region region-id="1846" left="282" top="4500" width="135.75" height="18" align-x="315" align-y="4512" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> 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значений</f> </text> </region> <region region-id="1853" left="90" top="4561.5" width="129.75" height="22.5" align-x="141.75" align-y="4578" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">KK</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>9642154.9240893759</ml:real> </result> </ml:eval> </math> <rendering item-idref="131"/> </region> <region region-id="1854" left="246" top="4566.75" width="62.25" height="17.25" align-x="286.5" align-y="4578" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> 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ruler-unit="cm"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="false" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="1876" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="1735" left="30" top="11.25" width="408" height="15.75" align-x="30" align-y="24" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="14"> 2.3. Метод, использующий разложение решения в ряд Тейлора. </f> </text> </region> <region region-id="1736" left="30" top="36.75" width="459" height="148.5" align-x="30" align-y="48" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr underline="false">Его можно также назвать методом припасовывания для решения краевых задач ОДУ.<sp count="2"/>Отличие этого метода от<sp count="2"/>предыдущих состоит в том, что исходное дифференциальное уравнение не надо<sp count="2"/>представлять в разностном виде. Искомое решение разбивается на ряд участков (число их в пределе должно стремиться к бесконечности), на каждом из которых оно представлено рядом Тейлора. Далее эти участки припасовываются (сшиваются). Алгоритм припасовывания описывается системой линейных алгебраических уравнений, имеющей, как правило, сильно разреженную матрицу. Именно эта матрица определяет пересчет краевых условий для каждого из участков. Решение же самой системы линейных алгебраических уравнений можно проводить методом Гаусса или другими методами, являющимися частными случаями последнего.</inlineAttr> </f> </text> </region> <region region-id="1406" left="24" top="192.75" width="447.75" height="54" align-x="24" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">По-прежнему будем рассматривать<sp count="2"/>систему линейных алгебраических уравнений</f> <f family="Arial Cyr" charset="204" size="12">с сильно разреженной матрицей, </f> <f family="Arial Cyr" charset="204" size="12">задающую алгоритм пересчета краевых условий</f> <f family="Arial Cyr" charset="204" size="12">от предыдущего участка к последующему<sp count="2"/>в общем виде и представляемую </f> <f family="Arial Cyr" charset="204" size="12">формулой (1)</f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="1413" left="36" top="258.75" width="276.75" height="17.25" align-x="75" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">B</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:id xml:space="preserve" font="1">...</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">D</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">m</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">C</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="2"/> </region> <region region-id="1756" left="336" top="258.75" width="61.5" height="15.75" align-x="364.5" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:range xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">i</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">m</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> </math> <rendering item-idref="3"/> </region> <region region-id="1757" left="414" top="258.75" width="66" height="15.75" align-x="446.25" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:range xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">k</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">K</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> </math> <rendering item-idref="4"/> </region> <region region-id="1758" left="492" top="258.75" width="15.75" height="15.75" align-x="498.75" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </ml:parens> </math> <rendering item-idref="5"/> </region> <region region-id="1486" left="24" top="282.75" width="491.25" height="196.5" align-x="24" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Где m - порядок диф. ур.; К - число участков припасовывания или дискретных значений переменных, вводимое для численного решения задачи; i - порядок производной или номер краевого условия; </f> <f family="Arial Cyr" charset="204" size="12">k - номер участка припасовывания. </f> <region region-id="1417" left="467.25" top="311.25" width="21" height="17.25" align-x="477" align-y="322.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </math> <rendering item-idref="6"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> - дискретные значения искомого решения, записываемые в матрицу "Z". Смысл матриц А, В, ..., С будет пояснен ниже. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">В частном случае, когда заданы все<sp count="2"/>Z</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> i,0 </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> - начальные условия задачи Коши, формула (1) определяет их пересчет (алгоритм пересчета м.б. разным) в конечные значения Z(x) на участке припасовывания (или начальные значения последующего участка припасовывания). Вместо начальных условий могут быть заданы краевые условия. Отличие от начальных условий состоит в том, что какая-то часть из них задана вначале, а другая - в конце интервала изменения переменных. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> Ниже приводится алгоритм решения для случая 4-х диагональной матрицы. Надо сказать, что методы "стрельбы", "прогонки" являются частными случаями метода Гаусса. Параметры дифференциального уравнения следующие: m=2,<sp count="2"/>далее см. по тексту. </inlineAttr> </f> </text> </region> <region region-id="1428" left="18" top="486.75" width="500.25" height="93" align-x="18" align-y="498" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Описанный ниже алгоритм справедлив для дифференциального<sp count="2"/>уравнения (линей-</f> <f family="Arial Cyr" charset="204" size="12">ного): </f> <region region-id="1427" left="51.75" top="501" width="87" height="15.75" align-x="111.75" align-y="512.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">z''</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">P</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="7"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>с заданными граничными условиями:<sp count="2"/> </inlineAttr> </f> <region region-id="1429" left="353.25" top="501" width="71.25" height="17.25" align-x="392.25" align-y="512.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">Z</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> </math> <rendering item-idref="8"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>,<sp count="3"/>Z(x</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> K </f> <f family="Arial Cyr" charset="204" size="12">)=Z</f> <f family="Arial Cyr" charset="204" size="12">0,K </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12">или<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">Z'(x</f> <f family="Arial Cyr" charset="204" size="12"> K </f> <f family="Arial Cyr" charset="204" size="12">)=Z</f> <f family="Arial Cyr" charset="204" size="12"> 1 </f> <f family="Arial Cyr" charset="204" size="12"> ,K </f> <f family="Arial Cyr" charset="204" size="12">), но просто обобщается на случай уравнений более высокого порядка в том числе нелинейных, имеющих гладкое решение и разрешенных относительно стар-</f> <f family="Arial Cyr" charset="204" size="12">шей производной. </f> <f family="Arial Cyr" charset="204" size="12">Для участка с номером "k" справедливы следующие соотношения:</f> </text> </region> <region region-id="1431" left="24" top="591" width="429" height="35.25" align-x="69" align-y="612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:parens> <ml:id xml:space="preserve">x</ml:id> </ml:parens> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:sequence> <ml:real>2</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:sequence> <ml:real>3</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:parens> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:sequence> <ml:real>4</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>4</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:parens> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve" font="1">...</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="9"/> </region> <region region-id="1455" left="492" top="600.75" width="15.75" height="15.75" align-x="498.75" align-y="612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </ml:parens> </math> <rendering item-idref="10"/> </region> <region region-id="1784" left="90" top="636" width="179.25" height="33.75" align-x="90" align-y="636" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="11"/> <rendering item-idref="12"/> </region> <region region-id="1783" left="30" top="648.75" width="21" height="13.5" align-x="30" align-y="660" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Или</f> </text> </region> <region region-id="1787" left="24" top="708.75" width="482.25" height="33" align-x="24" align-y="720" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Здесь в (2) введены следующие обозначения: Z</f> <f family="Arial Cyr" charset="204" size="12"> 0,k </f> <f family="Arial Cyr" charset="204" size="12">(x) - функция от х;<sp count="2"/>z</f> <f family="Arial Cyr" charset="204" size="12"> i,k </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">/</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">i! - коэффициент</f> <f family="Arial Cyr" charset="204" size="12">ряда Тейлора функции </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> 0,k </f> <f family="Arial Cyr" charset="204" size="12">(x). Отсюда из уравнения (2) формально можно записать</f> </text> </region> <region region-id="1792" left="36" top="762" width="204.75" height="33" align-x="36" align-y="762" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="13"/> <rendering item-idref="14"/> </region> <region region-id="1793" left="30" top="810.75" width="393.75" height="16.5" align-x="30" align-y="822" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">и получить значение первой производной от </f> <f family="Arial Cyr" charset="204" size="12">функции </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> 0,k </f> <f family="Arial Cyr" charset="204" size="12">(x) на участке k</f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> </text> </region> <region region-id="1434" left="24" top="843" width="349.5" height="35.25" align-x="70.5" align-y="864" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:parens> <ml:id xml:space="preserve">x</ml:id> </ml:parens> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:sequence> <ml:real>2</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:sequence> <ml:real>3</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:sequence> <ml:real>4</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:parens> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve" font="1">...</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="15"/> </region> <region region-id="1454" left="492" top="852.75" width="15.75" height="15.75" align-x="498.75" align-y="864" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>3</ml:real> </ml:parens> </math> <rendering item-idref="16"/> </region> <region region-id="1794" left="30" top="882" width="477" height="59.25" align-x="30" align-y="894" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Именно эти выражения (2)-(3) необходимо пересчитывать на каждом участке </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12"> из </f> <f family="Arial Cyr" charset="204" size="12">начальных<sp count="2"/>Z</f> <f family="Arial Cyr" charset="204" size="12"> i,k </f> <f family="Arial Cyr" charset="204" size="12">в конечные</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> i,k+1 </f> <f family="Arial Cyr" charset="204" size="12">, где<sp count="2"/>i=0,1,...(n-1),<sp count="3"/> </f> <f family="Arial Cyr" charset="204" size="12">n</f> <f family="Arial Cyr" charset="204" size="12"> - порядок дифференциального</f> <f family="Arial Cyr" charset="204" size="12">уравнения или число задаваемых краевых условий. Коротко первую производную</f> <f family="Arial Cyr" charset="204" size="12">на участке<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12"> можно записать так:</f> </text> </region> <region region-id="1795" left="156" top="956.25" width="116.25" height="37.5" align-x="156" align-y="978" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <region region-id="1724" left="156" top="957" width="119.25" height="36.75" align-x="202.5" align-y="978" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:parens> <ml:id xml:space="preserve">x</ml:id> </ml:parens> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:parens> <ml:id xml:space="preserve">x</ml:id> </ml:parens> </ml:apply> </ml:parens> </ml:lambda> </ml:apply> </ml:apply> </math> <rendering item-idref="17"/> </region> </text> </region> <region region-id="1796" left="36" top="996.75" width="474" height="46.5" align-x="36" align-y="1008" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Значения коэффициентов ряда Тейлора функции </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> 0,k </f> <f family="Arial Cyr" charset="204" size="12">(x) (2) определяются из началь- ных (краевых) условий и путем дифференцирования исходного уравнения с исклю- чением старшей производной и подстановкой x=x</f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12">как показано ниже</f> </text> </region> <region region-id="1438" left="90" top="1050.75" width="102" height="17.25" align-x="121.5" align-y="1062" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>2</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="18"/> </region> <region region-id="1439" left="216" top="1050.75" width="159" height="17.25" align-x="247.5" align-y="1062" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>3</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f'</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P'</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="19"/> </region> <region region-id="1453" left="492" top="1056.75" width="15.75" height="15.75" align-x="498.75" align-y="1068" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>4</ml:real> </ml:parens> </math> <rendering item-idref="20"/> </region> <region region-id="1442" left="96" top="1086.75" width="275.25" height="17.25" align-x="127.5" align-y="1098" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>4</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f''</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P''</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P'</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="21"/> </region> <region region-id="1458" left="492" top="1086.75" width="15.75" height="15.75" align-x="498.75" align-y="1098" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>5</ml:real> </ml:parens> </math> <rendering item-idref="22"/> </region> <region region-id="1800" left="108" top="1122" width="125.25" height="33" align-x="108" align-y="1122" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="23"/> <rendering item-idref="24"/> </region> <region region-id="1801" left="30" top="1134.75" width="45" height="13.5" align-x="30" align-y="1146" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Причем:</f> </text> </region> <region region-id="1464" left="18" top="1170" width="485.25" height="36" align-x="18" align-y="1182" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Учитывая. что<sp count="2"/> </f> <region region-id="1462" left="104.25" top="1170.75" width="69" height="17.25" align-x="134.25" align-y="1182" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="25"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> , Z</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> i,k </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">(x</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> k+1 </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">)=Z</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">i,k+1</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="3"/>и, подставляя в (2), (3) выражения</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> (4), (5), получим (для k=0..K-1):</inlineAttr> </f> </text> </region> <region region-id="1465" left="30" top="1218.75" width="174" height="17.25" align-x="72.75" align-y="1230" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">B</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">C</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="26"/> </region> <region region-id="1495" left="222" top="1218.75" width="174" height="17.25" align-x="264.75" align-y="1230" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">D</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">E</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">G</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="27"/> </region> <region region-id="1496" left="408" top="1218.75" width="66" height="15.75" align-x="440.25" align-y="1230" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:range xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">k</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">K</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> </math> <rendering item-idref="28"/> </region> <region region-id="1470" left="492" top="1218.75" width="15.75" height="15.75" align-x="498.75" align-y="1230" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>6</ml:real> </ml:parens> </math> <rendering item-idref="29"/> </region> <region region-id="1725" left="30" top="1240.5" width="420.75" height="79.5" align-x="30" align-y="1266" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Где:<sp count="2"/> </f> <region region-id="1726" left="60" top="1241.25" width="229.5" height="39" align-x="81.75" align-y="1266" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P'</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P''</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="30"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>,<sp count="2"/> </inlineAttr> </f> <region region-id="1728" left="306.75" top="1241.25" width="130.5" height="39" align-x="327" align-y="1266" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">B</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P'</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="31"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> , </inlineAttr> </f> <region region-id="1729" left="30" top="1281" width="194.25" height="39" align-x="51" align-y="1305.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">C</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f'</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f''</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="32"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ,<sp count="2"/> </inlineAttr> </f> <region region-id="1731" left="243" top="1284.75" width="57" height="36.75" align-x="264" align-y="1305.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">D</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:lambda> </ml:apply> </ml:apply> </math> <rendering item-idref="33"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> , </f> </inlineAttr> <region region-id="1732" left="312.75" top="1284.75" width="54.75" height="36.75" align-x="332.25" align-y="1305.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">E</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">B</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:lambda> </ml:apply> </ml:apply> </math> <rendering item-idref="34"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> ,<sp count="2"/> </f> </inlineAttr> <region region-id="1733" left="384" top="1284.75" width="57" height="36.75" align-x="405" align-y="1305.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">G</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">C</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:lambda> </ml:apply> </ml:apply> </math> <rendering item-idref="35"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/> </inlineAttr> </f> </text> </region> <region region-id="1473" left="24" top="1326.75" width="491.25" height="54" align-x="24" align-y="1338" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Здесь коэффициенты A, B, C, D, E, G известны и будут определяться таким образом ниже при решении конкретных задач. Формулы (6) далее будут использованы как рекуррентные при нахождении решения краевой задачи. Они представляют собой систему линейных алгебраических уравнений в виде 4-х диагональной матрицы. </f> </text> </region> <region region-id="1867" left="30" top="1410.75" width="420" height="27" align-x="30" align-y="1422" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Решать (6) можно методом Гаусса, последовательно исключая неизвестные переменные.</f> </text> </region> <region region-id="1859" left="30" top="1452.75" width="461.25" height="63" align-x="30" align-y="1464" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">В первой паре уравнений (6)<sp count="2"/>известным будет Z</f> <f family="Arial Cyr" charset="204" size="12"> 0,0 </f> <f family="Arial Cyr" charset="204" size="12"> . Следовательно,<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">значения</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> 0,1<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">и</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> 1 </f> <f family="Arial Cyr" charset="204" size="12"> ,1 </f> <f family="Arial Cyr" charset="204" size="12">можно будет выразить через </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> 1 </f> <f family="Arial Cyr" charset="204" size="12"> ,0 </f> <f family="Arial Cyr" charset="204" size="12">. Этот процесс исключения </f> <f family="Arial Cyr" charset="204" size="12">можно продолжить для второй и последующих пар и всякий раз выражать </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12"> ,k<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">через</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> 1 </f> <f family="Arial Cyr" charset="204" size="12"> ,0 </f> <f family="Arial Cyr" charset="204" size="12">. Предположим это нам удалось сделать для<sp count="2"/>(k-1)-й пары<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">уравнений: </f> </text> </region> <region region-id="1474" left="54" top="1530" width="156.75" height="18" align-x="94.5" align-y="1542" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="36"/> </region> <region region-id="1476" left="246" top="1530.75" width="41.25" height="15.75" align-x="275.25" align-y="1542" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:range xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">i</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:range> </math> <rendering item-idref="37"/> </region> <region region-id="1860" left="480" top="1530.75" width="15.75" height="15.75" align-x="486.75" align-y="1542" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>7</ml:real> </ml:parens> </math> <rendering item-idref="38"/> </region> <region region-id="1481" left="24" top="1553.25" width="447" height="127.5" align-x="24" align-y="1566" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Тогда, подставляя (7) в (6), получим: </f> <region region-id="1482" left="234" top="1554" width="123" height="18" align-x="263.25" align-y="1566" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="39"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> . Сравнивая результат подстановки с последним выражением, получим рекуррентные формулы вычисления коэффициентов </inlineAttr> </f> <region region-id="1483" left="244.5" top="1586.25" width="24" height="16.5" align-x="255.75" align-y="1598.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">α</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:sequence> </math> <rendering item-idref="40"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> (всё это справедливо для линейных </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">уравнений)</inlineAttr> </f> <region region-id="1572" left="24" top="1630.5" width="185.25" height="18" align-x="57" align-y="1642.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="41"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ,<sp count="3"/> </inlineAttr> </f> <region region-id="1573" left="225" top="1630.5" width="221.25" height="18" align-x="257.25" align-y="1642.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="42"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ,</inlineAttr> </f> <region region-id="1574" left="24" top="1649.25" width="184.5" height="18" align-x="57" align-y="1661.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">E</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="43"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/>,<sp count="3"/> </f> </inlineAttr> <region region-id="1575" left="228" top="1649.25" width="220.5" height="18" align-x="260.25" align-y="1661.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">E</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">G</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="44"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">Ниже будет приведен алгоритм вычисления этих </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12">коэффициентов.</f> </text> </region> <region region-id="1802" left="492" top="1626.75" width="21" height="15.75" align-x="501.75" align-y="1638" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:mult style="auto-select"/> <ml:real>7</ml:real> <ml:id xml:space="preserve">a</ml:id> </ml:apply> </ml:parens> </math> <rendering item-idref="45"/> </region> <region region-id="1354" left="24" top="1680.75" width="487.5" height="73.5" align-x="24" align-y="1698" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Приведем</f> <f family="Arial Cyr" charset="204" size="12"> пример </f> <f family="Arial Cyr" charset="204" size="12"> решения уравнения </f> <region region-id="49" left="247.5" top="1681.5" width="60.75" height="21" align-x="282.75" align-y="1698" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">z''</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="46"/> </region> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/>с граничными условиями: </f> <region region-id="54" left="462.75" top="1686.75" width="42.75" height="15.75" align-x="490.5" align-y="1698" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="47"/> </region> <f family="Arial Cyr" charset="204" size="12"> ,<sp count="2"/> </f> <region region-id="55" left="33.75" top="1703.25" width="52.5" height="34.5" align-x="71.25" align-y="1723.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> </math> <rendering item-idref="48"/> </region> <f family="Arial Cyr" charset="204" size="12"> <sp count="3"/>(другим примером может быть z'' + p(t)*z = f(t), где p(t)=t^2 или t, </f> <f family="Arial Cyr" charset="204" size="12">f(t)=sin(t)*e</f> <f family="Arial Cyr" charset="204" size="12">-t</f> <f family="Arial Cyr" charset="204" size="12">).</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="3"/>Точное решение этого уравнения есть:</f> </text> </region> <region region-id="1577" left="24" top="1779" width="230.25" height="47.25" align-x="53.25" align-y="1812" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">yt</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="49"/> </region> <region region-id="1580" left="336" top="1800.75" width="53.25" height="15.75" align-x="367.5" align-y="1812" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:provenance expr-id="1"> <originRef doc-id="66f65496-debc-41ba-835b-b3474937ab71" version-id="1c6b8a94-b7d0-4499-84b2-8f948851f9b2" branch-id="00000000-0000-0000-0000-000000000000" revision-num="25" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЧислМетоды\КраевыеЗадачи\МетСтрельбы.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="66f65496-debc-41ba-835b-b3474937ab71" version-id="1c6b8a94-b7d0-4499-84b2-8f948851f9b2" branch-id="00000000-0000-0000-0000-000000000000" revision-num="25" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЧислМетоды\КраевыеЗадачи\МетСтрельбы.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">72f877ffa76c567a47fae3ce36224632</contentHash> <ml:apply> <ml:id xml:space="preserve">yt</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:provenance> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="50"> <element-image-map> <box left="1.5" top="0.75" width="23.25" height="14.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="1581" left="414" top="1791.75" width="63" height="34.5" align-x="455.25" align-y="1812" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">yt</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </result> </ml:eval> </math> <rendering item-idref="51"/> </region> <region region-id="1872" left="30" top="1848" width="89.25" height="14.25" align-x="30" align-y="1860" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Пусть заданы: </f> </text> </region> <region region-id="1868" left="144" top="1848.75" width="35.25" height="17.25" align-x="162.75" align-y="1860" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define 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<ml:real>0</ml:real> <ml:real>2</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="58"/> </region> <region region-id="1424" left="36" top="1923" width="56.25" height="35.25" align-x="49.5" align-y="1944" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">b</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">KK</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="59"/> </region> <region region-id="1423" left="114" top="1927.5" width="99" height="21" align-x="126.75" align-y="1944" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math 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<math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">a</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="64"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <region region-id="1502" left="193.5" top="1974.75" width="48.75" height="17.25" align-x="226.5" align-y="1986" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:id xml:space="preserve">a</ml:id> </ml:define> </math> <rendering item-idref="65"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ,<sp count="2"/> </inlineAttr> </f> <region region-id="1503" left="257.25" top="1974.75" width="57.75" height="17.25" align-x="298.5" align-y="1986" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <ml:real>2</ml:real> </ml:define> </math> <rendering item-idref="66"/> </region> </text> </region> <region region-id="1875" left="30" top="2004.75" width="368.25" height="13.5" align-x="30" align-y="2016" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">3)<sp count="2"/>дискретные значения функций, вычисленные в 3*(КК+1) - точках.</f> </text> </region> <region region-id="1295" left="42" top="2028.75" width="39" height="17.25" align-x="48" align-y="2040" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">p</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> </math> <rendering item-idref="67"/> </region> <region region-id="1297" left="96" top="2028.75" width="36.75" height="17.25" align-x="99.75" align-y="2040" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> </math> <rendering item-idref="68"/> </region> <region region-id="1300" left="30" top="2058.75" width="462" height="13.5" align-x="30" align-y="2070" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">4)<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">дискретные значения коэффициентов, вычисленные в</f> <f family="Arial Cyr" charset="204" size="12"> (КК+1) - точках:</f> </text> </region> <region region-id="1275" left="24" top="2117.25" 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xml:space="preserve">h</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">AA</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:define> </math> <rendering item-idref="70"/> </region> <region region-id="1278" left="18" top="2183.25" width="361.5" height="39" align-x="69.75" align-y="2208" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">CC</ml:id> <ml:boundVars> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:real>2</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">p</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="71"/> </region> <region region-id="1279" left="396" top="2187" width="120" height="36.75" align-x="448.5" align-y="2208" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">GG</ml:id> <ml:boundVars> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">CC</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:define> </math> <rendering item-idref="72"/> </region> <region region-id="1280" left="18" top="2243.25" width="207.75" height="39" align-x="60.75" align-y="2268" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">B</ml:id> <ml:boundVars> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">p</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:id xml:space="preserve">p</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:factorial/> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="73"/> </region> <region region-id="1281" left="258" top="2247" width="100.5" height="36.75" align-x="300" align-y="2268" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">E</ml:id> <ml:boundVars> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">B</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:define> </math> <rendering item-idref="74"/> </region> <region region-id="1366" left="18" top="2297.25" width="468.75" height="51" align-x="18" align-y="2310" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Зададим также для единообразия: </f> <region region-id="1303" left="216.75" top="2298" width="50.25" height="18" align-x="250.5" align-y="2310" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="75"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>, </inlineAttr> </f> <region region-id="1304" left="280.5" top="2298" width="50.25" height="18" align-x="314.25" align-y="2310" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="76"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ,<sp count="2"/> </inlineAttr> </f> <region region-id="1306" left="344.25" top="2298" width="48.75" height="18" align-x="377.25" align-y="2310" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:id xml:space="preserve">a</ml:id> </ml:define> </math> <rendering item-idref="77"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> , </inlineAttr> </f> <region region-id="1307" left="402.75" top="2298" width="49.5" height="18" align-x="435.75" align-y="2310" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="78"/> </region> <f family="Arial Cyr" charset="204" size="12">Т</f> <f family="Arial Cyr" charset="204" size="12">аким образом, получили:</f> <f family="Arial Cyr" charset="204"> <sp/> </f> <region region-id="1511" left="169.5" top="2316.75" width="133.5" height="18" align-x="201.75" align-y="2328.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="79"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/>,<sp count="2"/> </f> </inlineAttr> <region region-id="1512" left="319.5" top="2316.75" width="133.5" height="18" align-x="351.75" align-y="2328.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="80"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> ,<sp count="3"/> </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12">и вычислим:</f> </text> </region> <region region-id="1365" left="36" top="2364" width="277.5" height="130.5" align-x="53.25" align-y="2376" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">α</ml:id> <ml:program> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:real>0</ml:real> </ml:localDefine> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:real>1</ml:real> </ml:localDefine> <ml:for> <ml:id xml:space="preserve">k</ml:id> <ml:range> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">KK</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> <ml:program> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">AA</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">B</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:localDefine> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">D</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:localDefine> </ml:program> </ml:for> <ml:id xml:space="preserve">α</ml:id> </ml:program> </ml:define> </math> <rendering item-idref="81"/> </region> <region region-id="1695" left="36" top="2520" width="331.5" height="130.5" align-x="52.5" align-y="2532" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">β</ml:id> <ml:program> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:id xml:space="preserve">a</ml:id> </ml:localDefine> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:real>0</ml:real> </ml:localDefine> <ml:for> <ml:id xml:space="preserve">k</ml:id> <ml:range> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">KK</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> <ml:program> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">AA</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">B</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">CC</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:localDefine> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">D</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">GG</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:localDefine> </ml:program> </ml:for> <ml:id xml:space="preserve">β</ml:id> </ml:program> </ml:define> </math> <rendering item-idref="82"/> </region> <region region-id="1589" left="36" top="2670.75" width="373.5" height="13.5" align-x="36" align-y="2682" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Для визуальной проверки выведем на дисплей некоторые значения:</f> </text> </region> <region region-id="1764" left="36" top="2700" width="89.25" height="18" align-x="76.5" align-y="2712" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.60393978861178022</ml:real> </result> </ml:eval> </math> <rendering item-idref="83"/> </region> <region region-id="1763" left="126" top="2700" width="96" height="18" align-x="166.5" align-y="2712" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.6039397877543341</ml:real> </result> </ml:eval> </math> <rendering item-idref="84"/> </region> <region region-id="1761" left="252" top="2700" width="63" height="18" align-x="293.25" align-y="2712" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.0000000000908043</ml:real> </result> </ml:eval> </math> <rendering item-idref="85"/> </region> <region region-id="1765" left="336" top="2695.5" width="126.75" height="22.5" align-x="377.25" align-y="2712" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2.6129656402994195E-10</ml:real> </result> </ml:eval> </math> <rendering item-idref="86"/> </region> <region region-id="1876" left="24" top="2729.25" width="490.5" height="65.25" align-x="24" align-y="2742" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Заметим, что коэффициенты </f> <region region-id="1712" left="191.25" top="2730" width="53.25" height="18" align-x="216" align-y="2742" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:sequence> </math> <rendering item-idref="87"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>, в данном примере не зависят от краевых (граничных) условий. Коэффициент </inlineAttr> </f> <region region-id="1713" left="228" top="2748.75" width="25.5" height="18" align-x="240" align-y="2760.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </math> <rendering item-idref="88"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> , зависит от Y</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> 0,0 </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> таким образом: при </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">Y</f> <f family="Arial Cyr" charset="204" size="12">0,0</f> <f family="Arial Cyr" charset="204" size="12">=0</f> <region region-id="1715" left="24" top="2772" width="82.5" height="18" align-x="64.5" align-y="2784" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <ml:real>0.3961</ml:real> </ml:apply> </math> <rendering item-idref="89"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">, при </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12">Y</f> <f family="Arial Cyr" charset="204" size="12">0,0</f> <f family="Arial Cyr" charset="204" size="12">=1 </f> <region region-id="1716" left="176.25" top="2772" width="101.25" height="18" align-x="216.75" align-y="2784" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:minus/> <ml:real>0.3961</ml:real> <ml:real>1</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="90"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ,<sp count="2"/>при </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">Y</f> <f family="Arial Cyr" charset="204" size="12">0,0</f> <f family="Arial Cyr" charset="204" size="12">=2<sp count="2"/> </f> <region region-id="1717" left="357" top="2772" width="101.25" height="18" align-x="397.5" align-y="2784" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:minus/> <ml:real>0.3961</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="91"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> и<sp count="2"/>т. д.</inlineAttr> </f> </text> </region> <region region-id="1770" left="36" top="2814.75" width="183.75" height="13.5" align-x="36" align-y="2826" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Напомним граничные условия:</f> </text> </region> <region region-id="1769" left="234" top="2814.75" width="75" height="17.25" align-x="260.25" align-y="2826" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">KK</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.5707963267948966</ml:real> </result> </ml:eval> </math> <rendering item-idref="92"/> </region> <region region-id="1768" left="318" top="2814.75" width="54" height="17.25" align-x="350.25" align-y="2826" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="93"/> </region> <region region-id="1767" left="384" top="2814.75" width="62.25" height="17.25" align-x="424.5" align-y="2826" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </result> </ml:eval> </math> <rendering item-idref="94"/> </region> <region region-id="1622" left="36" top="2886.75" width="295.5" height="16.5" align-x="36" align-y="2898" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Исходя из (7), если задана функция Y</f> <f family="Arial Cyr" charset="204" size="12"> 0,KK </f> <f family="Arial Cyr" charset="204" size="12">, </f> <f family="Arial Cyr" charset="204" size="12"> вычисляем:</f> </text> </region> <region region-id="1804" left="360" top="2876.25" width="125.25" height="38.25" align-x="393" align-y="2898" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="95"/> </region> <region region-id="1631" left="36" top="2929.5" width="437.25" height="48.75" align-x="36" align-y="2952" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Если же задана её производная Y</f> <f family="Arial Cyr" charset="204" size="12"> 1,KK </f> <f family="Arial Cyr" charset="204" size="12">, то вычисляем:<sp count="3"/> </f> <region region-id="1632" left="345.75" top="2930.25" width="126.75" height="38.25" align-x="378.75" align-y="2952" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="96"/> </region> </text> </region> <region region-id="1648" left="30" top="2994.75" width="94.5" height="13.5" align-x="30" align-y="3006" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Вычислим далее:</f> </text> </region> <region region-id="1649" left="138" top="2994.75" width="58.5" height="15.75" align-x="152.25" align-y="3006" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">k</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="97"/> </region> <region region-id="1650" left="210" top="2994" width="135.75" height="18" align-x="243" align-y="3006" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="98"/> </region> <region region-id="1651" left="360" top="2994" width="135.75" height="18" align-x="393" align-y="3006" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="99"/> </region> <region region-id="1647" left="30" top="3024.75" width="180.75" height="13.5" align-x="30" align-y="3036" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Выведем на дисплей значение:</f> </text> </region> <region region-id="1745" left="216" top="3024.75" width="96" height="17.25" align-x="256.5" align-y="3036" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.60393978738954834</ml:real> </result> </ml:eval> </math> <rendering item-idref="100"/> </region> <region region-id="1771" left="30" top="3054.75" width="474.75" height="27" align-x="30" align-y="3066" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Остальные компоненты представлены в виде разворачиваемой в электронной версии пособия матрицы Y.</f> </text> </region> <region region-id="1652" left="18" top="3125.25" width="465.75" height="39.75" align-x="33.75" align-y="3144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Y</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="2" cols="1001"> <ml:real>1</ml:real> <ml:real>1.3960602112614515</ml:real> <ml:real>1.0021929247043186</ml:real> <ml:real>1.3960572558873</ml:real> <ml:real>1.0043858401255794</ml:real> <ml:real>1.3960483923522911</ml:real> <ml:real>1.006578736986318</ml:real> <ml:real>1.3960336245449576</ml:real> <ml:real>1.0087716060151841</ml:real> 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<ml:real>2.0037592743154544</ml:real> <ml:real>-0.5926718274883308</ml:real> <ml:real>2.0028260938727591</ml:real> <ml:real>-0.59549025617531359</ml:real> <ml:real>2.0018884870008642</ml:real> <ml:real>-0.59830773089437406</ml:real> <ml:real>2.0009464552043532</ml:real> <ml:real>-0.60112424388480667</ml:real> <ml:real>2</ml:real> <ml:real>-0.60393978738954834</ml:real> </ml:matrix> </result> </ml:eval> <resultFormat> <table item-idref="101"/> </resultFormat> </math> <rendering item-idref="102"/> </region> <region region-id="1772" left="42" top="3192.75" width="405.75" height="13.5" align-x="42" align-y="3204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Полученные приближенное и точное решения можно сравнить на графике:</f> </text> </region> <region region-id="1773" left="42" top="3264" width="255" height="135.75" align-x="42" align-y="3264" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="103"/> <rendering item-idref="104"/> </region> <region region-id="1776" left="390" top="3312.75" width="77.25" height="17.25" align-x="418.5" align-y="3324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:real>700</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.0995574287564276</ml:real> </result> </ml:eval> </math> <rendering item-idref="105"/> </region> <region region-id="1775" left="384" top="3354.75" width="91.5" height="17.25" align-x="426.75" align-y="3366" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>700</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2.0829064863423685</ml:real> </result> </ml:eval> </math> <rendering item-idref="106"/> </region> <region region-id="1746" left="42" top="3456" width="465" height="27.75" align-x="42" align-y="3468" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12"> Упражнение. </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false"> Привести п</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false">ример </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false">других краевых условий, когда частного решения нет<sp count="8"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false">(</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false">М</inlineAttr> </f> <f family="Arial Cyr" charset="204">ожно взять граничные условия: y(0)=1, y'(Pi/2)=1).</f> </text> </region> <region region-id="1750" left="30" top="3510.75" width="463.5" height="27" align-x="30" align-y="3522" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Если Вы хотите сравнить полученное Вами решение с ответом, откройте в электрон-</f> <f family="Arial Cyr" charset="204" size="12">ной версии пособия ниже представленную закрытую область.</f> </text> </region> <region region-id="1753" left="0" top="3543" width="6008.25" align-x="8.25" align-y="3552" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="" height="1388.25"> <area is-collapsed="false" name="" show-name="false" show-border="true" show-icon="true" show-timestamp="true" allow-expand="false" is-locked="false" timestamp="" top-lock-id="1753" bottom-lock-id="1754" bottom-tag=""> <region region-id="1864" left="30" top="3561.75" width="470.25" height="16.5" align-x="30" align-y="3576" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">По-прежнему рассматриваем уравнение y'' + y = e</f> <f family="Arial Cyr" charset="204" size="12"> -x </f> <sp/> <f family="Arial Cyr" charset="204" size="12">и для заданных граничных условий</f> </text> </region> <region region-id="1806" left="24" top="3591.75" width="489.75" height="136.5" align-x="24" align-y="3612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0" size="10"> <inlineAttr font-style="normal" underline="false"> y(0)=1, y'(Pi/2)=1 </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false">его решение будет</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-style="normal" underline="false">: </inlineAttr> </f> <region region-id="1538" left="215.25" top="3592.5" width="179.25" height="33.75" align-x="244.5" align-y="3612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="107"/> </region> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">Где не</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">известный коэффициент В определяется из 2-го граничного условия:</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="10"> <sp/> </f> </inlineAttr> <region region-id="1539" left="24" top="3654" width="183.75" height="33.75" align-x="54" align-y="3673.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">y'</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:neg/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="108"/> </region> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">,<sp count="2"/> </inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">то есть</f> </inlineAttr> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <region region-id="1540" left="266.25" top="3640.5" width="145.5" height="47.25" align-x="306.75" align-y="3673.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">y'</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:lessThan/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>-1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:real>0</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="109"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="10">.</f> </inlineAttr> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">Это означает, что</f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, </inlineAttr> </f> <f family="Arial" charset="0">y'(Pi/2) <>1! </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false">Следовательно</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="10"> <inlineAttr font-style="normal" underline="false">, </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" underline="false">какой бы</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-style="normal" underline="false"> <sp/> </inlineAttr> </f> <inlineAttr font-style="normal" underline="false"> <f family="Arial Cyr" charset="204" size="12">коэффициент В мы не выбирали, полученное таким образом решение решение заданным краевым условиям удовлетворять не будет. Представим решение как функцию параметра В</f> </inlineAttr> </text> </region> <region region-id="1554" left="30" top="3790.5" width="192" height="33.75" align-x="69.75" align-y="3810" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">y</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">B</ml:id> </ml:boundVars> </ml:function> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:define> </math> <rendering item-idref="110"/> </region> <region region-id="1553" left="306" top="3777" width="137.25" height="47.25" align-x="387.75" align-y="3810" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>-1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.603939788175381</ml:real> </result> </ml:eval> </math> <rendering item-idref="111"/> </region> <region region-id="1808" left="30" top="3840.75" width="483" height="54" align-x="30" align-y="3852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">и построим графики y(t,B) для разных В. Все эти и другие кривые в точке t=pi/2, будут иметь одну и ту же касательную с угловым коэффициентом - 0,6039. Кроме этого у</f> <f family="Arial Cyr" charset="204" size="12">множества решений </f> <f family="Arial Cyr" charset="204" size="12">y(t,B) есть стационарная точка t=pi, через которую проходит каж-</f> <f family="Arial Cyr" charset="204" size="12">дая кривая этого семейства.</f> </text> </region> <region region-id="1556" left="36" top="3918" width="251.25" height="139.5" align-x="36" align-y="3918" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="112"/> <rendering item-idref="113"/> </region> <region region-id="1679" left="18" top="4092" width="490.5" height="31.5" align-x="18" align-y="4104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Сформулируем запись граничных условий</f> <f family="Arial" charset="0" size="12">: </f> <region region-id="1564" left="264" top="4092.75" width="49.5" height="17.25" align-x="297" align-y="4104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="114"/> </region> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ,<sp count="2"/> </inlineAttr> </f> <region region-id="1565" left="328.5" top="4092.75" width="57.75" height="17.25" align-x="369.75" align-y="4104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="115"/> </region> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">и найдем коэффициенты альфа и бетта для участка k=0</inlineAttr> </f> </text> </region> <region region-id="1826" left="24" top="4140" width="50.25" height="18" align-x="57.75" align-y="4152" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="116"/> </region> <region region-id="1827" left="102" top="4140" width="50.25" height="18" align-x="135.75" align-y="4152" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="117"/> </region> <region region-id="1828" left="180" top="4140" width="48.75" height="18" align-x="213" align-y="4152" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:id xml:space="preserve">a</ml:id> </ml:define> </math> <rendering item-idref="118"/> </region> <region region-id="1829" left="258" top="4140" width="49.5" height="18" align-x="291" align-y="4152" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="119"/> </region> <region region-id="1830" left="24" top="4164.75" width="422.25" height="13.5" align-x="24" align-y="4176" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">На участке k=0 решение и его производная будут определяться уравнениями</f> </text> </region> <region region-id="1831" left="18" top="4211.25" width="468.75" height="18.75" align-x="18" align-y="4224" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0" size="12"> <sp/> </f> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <region region-id="1570" left="24" top="4212" width="129.75" height="18" align-x="55.5" align-y="4224" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="120"/> </region> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/>,<sp count="4"/> </inlineAttr> </f> <region region-id="1571" left="177.75" top="4212" width="130.5" height="18" align-x="208.5" align-y="4224" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="121"/> </region> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/> </inlineAttr> </f> <f family="Arial" charset="0" size="12"> <sp/> </f> </text> </region> <region region-id="1832" left="24" top="4242.75" width="468" height="46.5" align-x="24" align-y="4254" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Так как Z</f> <f family="Arial Cyr" charset="204" size="12"> 1,0 </f> <f family="Arial Cyr" charset="204" size="12"> неизвестно на начальном этапе и задается только как граничное условие </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> 1,k </f> <f family="Arial Cyr" charset="204" size="12"> в конце траектории описывающей решение, то, используя уранение (7), для </f> <f family="Arial Cyr" charset="204" size="12">Z</f> <f family="Arial Cyr" charset="204" size="12"> 1,0 </f> <f family="Arial Cyr" charset="204" size="12">получим</f> </text> </region> <region region-id="1654" left="138" top="4316.25" width="125.25" height="38.25" align-x="171" align-y="4338" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="122"/> </region> <region region-id="1665" left="312" top="4321.5" width="110.25" height="22.5" align-x="344.25" align-y="4338" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>6138388362.3151617</ml:real> </result> </ml:eval> </math> <rendering item-idref="123"/> </region> <region region-id="1833" left="36" top="4368.75" width="434.25" height="27" align-x="36" align-y="4380" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Проведем визуализацию значений коэффициентов альфа и бетта, которые они</f> <f family="Arial Cyr" charset="204" size="12">принимают в конце интервала (траектории)</f> </text> </region> <region region-id="1841" left="42" top="4416" width="63" height="18" align-x="83.25" align-y="4428" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.0000000000908043</ml:real> </result> </ml:eval> </math> <rendering item-idref="124"/> </region> <region region-id="1840" left="120" top="4411.5" width="126.75" height="22.5" align-x="161.25" align-y="4428" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2.6129656402994195E-10</ml:real> </result> </ml:eval> </math> <rendering item-idref="125"/> </region> <region region-id="1839" left="264" top="4416" width="89.25" height="18" align-x="304.5" align-y="4428" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.60393978861178022</ml:real> </result> </ml:eval> </math> <rendering item-idref="126"/> </region> <region region-id="1838" left="366" top="4416" width="96" height="18" align-x="406.5" align-y="4428" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.6039397877543341</ml:real> </result> </ml:eval> </math> <rendering item-idref="127"/> </region> <region region-id="1847" left="30" top="4452.75" width="407.25" height="30" align-x="30" align-y="4464" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Также вычислим вектор решения {Y</f> <f family="Arial Cyr" charset="204" size="12"> 0,k </f> <f family="Arial Cyr" charset="204" size="12">}<sp count="2"/>и вектор его производной<sp count="2"/>{Y</f> <f family="Arial Cyr" charset="204" size="12"> 1,k </f> <f family="Arial Cyr" charset="204" size="12">} по </f> <f family="Arial Cyr" charset="204" size="12">уравнению (7)</f> </text> </region> <region region-id="1844" left="42" top="4500.75" width="58.5" height="15.75" align-x="56.25" align-y="4512" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">k</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">KK</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="128"/> </region> <region region-id="1845" left="132" top="4500" width="135.75" height="18" align-x="165" align-y="4512" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">α</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">β</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="129"/> </region> <region region-id="1846" left="282" top="4500" width="135.75" height="18" align-x="315" align-y="4512" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> 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значений</f> </text> </region> <region region-id="1853" left="90" top="4561.5" width="129.75" height="22.5" align-x="141.75" align-y="4578" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">KK</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>9642154.9240893759</ml:real> </result> </ml:eval> </math> <rendering item-idref="131"/> </region> <region region-id="1854" left="246" top="4566.75" width="62.25" height="17.25" align-x="286.5" align-y="4578" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> 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