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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="277" 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="1" left="42" top="13.5" width="407.25" height="51" align-x="42" 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="true"> <f family="Arial Cyr" charset="204" size="12"> Геометрическая интерпретация решения дифференциального уравнения </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <region region-id="2" left="103.5" top="27.75" width="84" height="36.75" align-x="147" align-y="48.75" 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:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:sequence> </ml:apply> </ml:apply> </math> <rendering item-idref="2"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">.</inlineAttr> </f> </text> </region> <region region-id="4" left="48" top="72" width="442.5" height="90" align-x="48" align-y="84" 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">Рассмотрим на примере уравнения с </f> <region region-id="5" left="260.25" top="72.75" width="78.75" height="15.75" align-x="297.75" align-y="84" show-border="false" show-highlight="true" 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">f</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="3"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Решением этого диф-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">ференциального уравнения будет семейство кривых </inlineAttr> </f> <region region-id="6" left="348.75" top="89.25" width="93.75" height="15.75" align-x="383.25" align-y="100.5" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math error="bad_dimensions Unitless Charge " optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">y1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:id xml:space="preserve">C</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="4"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, где</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">константа С определяется начальными условиями: x</f> </inlineAttr> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> o </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, y(x</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> o </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">)=y</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> o </inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">. </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">Функция f(x,y) задает поверхность, а y1(x), кривая под этой поверхностью, явля-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">ется решением </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">дифференциального уравнения, в каждой точке которого угловой коэффициент касательной равен значению f(x,y) в этой точке.</f> </text> </region> <region region-id="12" left="48" top="174.75" width="441.75" height="87.75" align-x="48" align-y="186" 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">Итак, зададим поверхность f(x,y) в виде дискретных точек и запишем её в</f> <f family="Arial Cyr" charset="204" size="12">матрицу М. Пусть </f> <region region-id="8" left="150" top="189.75" width="62.25" height="16.5" align-x="194.25" align-y="201.75" 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:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </math> <rendering item-idref="5"/> </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="9" left="216" top="190.5" width="57.75" height="15.75" align-x="258" align-y="201.75" 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:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:real>-2</ml:real> <ml:id xml:space="preserve">y</ml:id> </ml:apply> <ml:real>2</ml:real> </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">. В качестве дискретного шага по</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">переменным x и у выберем </inlineAttr> </f> <region region-id="13" left="205.5" top="207" width="45" height="15.75" align-x="219" align-y="218.25" show-border="false" show-highlight="true" 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:real>0.04</ml:real> </ml:define> </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"> (здесь он один и тот же, но может быть</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">разным и переменным)</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Введём обозначения: n - </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">номер строки матрицы </f> <f family="Arial Cyr" charset="204" size="12">М; </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">m - номер столбца матрицы М. Число строк и столбцов матрицы М, </f> </inlineAttr> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">определим как N:</f> </inlineAttr> </text> </region> <region region-id="52" left="60" top="273.75" width="90" height="34.5" align-x="83.25" align-y="294" 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">N1</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="76" left="180" top="282.75" width="56.25" height="15.75" align-x="202.5" align-y="294" 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">N1</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>157</ml:real> </result> </ml:eval> </math> <rendering item-idref="9"/> </region> <region region-id="75" left="252" top="282.75" width="54.75" height="15.75" align-x="265.5" align-y="294" 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">n</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">N1</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="277" left="60" top="322.5" width="78" height="33.75" align-x="83.25" 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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">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="97" left="390" top="366.75" width="75.75" height="17.25" align-x="411.75" align-y="378" 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:id xml:space="preserve">m</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:real>-1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="62" left="60" top="390.75" width="67.5" height="17.25" align-x="78" align-y="402" 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>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-3.1415926535897931</ml:real> </result> </ml:eval> </math> <rendering item-idref="16"/> </region> <region region-id="63" left="144" top="390.75" width="72.75" height="17.25" align-x="167.25" align-y="402" 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>78</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.021592653589793009</ml:real> </result> </ml:eval> </math> <rendering item-idref="17"/> </region> <region region-id="64" left="264" top="390.75" width="66" height="17.25" align-x="287.25" align-y="402" 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>79</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.018407346410207026</ml:real> </result> </ml:eval> </math> <rendering item-idref="18"/> </region> <region region-id="39" left="54" top="420" width="447.75" height="60" align-x="54" align-y="432" 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:<sp count="2"/> </f> <region region-id="41" left="96" top="420.75" width="93" height="17.25" align-x="134.25" align-y="432" 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">M</ml:id> <ml:sequence> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="19"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Так как f(x,y) не зависит от </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" underline="false" line-through="false">у</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, то у матрицы </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" underline="false" line-through="false"> М </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> все </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">столбцы будут одинаковы и задавать она будет цилиндрическую поверхность с образующей, параллельной оси ординат (0</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" underline="false" line-through="false">у</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">), а направляющая будет лежать в плоскости </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" underline="false" line-through="false">f</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">0</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="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">(аппликата-абсцисса)</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Теперь построим поверхность f(x,y) рис.1.</inlineAttr> </f> </text> </region> <region region-id="79" left="54" top="492" width="450.75" height="94.5" align-x="54" align-y="504" 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 матрицу решений </f> <f family="Arial Cyr" charset="204" size="12"> y </f> <f family="Arial Cyr" charset="204" size="12">1(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">)=sin(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">)+</f> <f family="Arial Cyr" charset="204" size="12"> C </f> <f family="Arial Cyr" charset="204" size="12">, обозначив её через </f> <f family="Arial Cyr" charset="204" size="12"> Y </f> <f family="Arial Cyr" charset="204" size="12">. </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">Пусть </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">С</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">=1. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">Зависимость </inlineAttr> </f> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12"> y </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">1(</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">x</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">) представим в дискретном виде и запишем в матрицу </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">Y</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">. Причем номер строки </inlineAttr> </f> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12"> n </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal"> матрицы будет абсциссой, а номер столбца </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">m</inlineAttr> </f> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12">=v</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">n</inlineAttr> </f> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12"> -<sp count="2"/>ординатой решения </f> </inlineAttr> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12">y</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">1(</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">x</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">). Сам же элемент матрицы </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">Y</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal"> n,m </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal"> будет представлять аппликату образующей цилиндрической поверхности с направляющей </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">y</inlineAttr> </f> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12">1(</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">x</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">).<sp count="2"/> </inlineAttr> </f> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12">Таким образом, зависимость </f> </inlineAttr> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12"> y </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">1(</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">x</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">) можно представить </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">вектором </inlineAttr> </f> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12"> v </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal"> n </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">.</inlineAttr> </f> </text> </region> <region region-id="121" left="60" top="604.5" width="75.75" height="33.75" align-x="82.5" align-y="624" 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">C1</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">FRAME</ml:id> <ml:real>10</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="122" left="180" top="603" width="148.5" height="35.25" align-x="198.75" align-y="624" 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">v</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">C1</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="123" left="360" top="612.75" width="45.75" height="17.25" align-x="378" align-y="624" 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">v</ml:id> <ml:real>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>24</ml:real> </result> </ml:eval> </math> <rendering item-idref="22"/> </region> <region region-id="124" left="438" top="612.75" width="48" height="17.25" align-x="464.25" align-y="624" 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">v</ml:id> <ml:id xml:space="preserve">N2</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </result> </ml:eval> </math> <rendering item-idref="23"/> </region> <region region-id="112" left="138" top="660.75" width="86.25" height="22.5" align-x="176.25" align-y="672" 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:id xml:space="preserve">n</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="77" left="60" top="750" width="429.75" height="326.25" align-x="273.75" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="25" disable-calc="false"> <inputs> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">M</ml:id> <ml:id xml:space="preserve">Y</ml:id> </ml:sequence> </inputs> <outputs/> </component> <rendering item-idref="26"/> </region> <region region-id="125" left="60" top="1098.75" width="458.25" height="40.5" align-x="60" align-y="1110" 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">Из рис.1 видно, что </f> <f family="Arial Cyr" charset="204" size="12">в каждой точке y1(x) угловой коэффициент касательной </f> <f family="Arial Cyr" charset="204" size="12">равен значению f(x,y) в этой точке. А поверхность f(x,y) является геометри-</f> <f family="Arial Cyr" charset="204" size="12">ческим местом таких точек. </f> </text> </region> <region region-id="158" left="66" top="1152" width="372" height="22.5" align-x="66" align-y="1164" 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">Убедимся в том, что v</f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12">h</f> <f family="Arial Cyr" charset="204" size="12">-1 есть синусоида: </f> <region region-id="159" left="300" top="1152.75" width="73.5" height="17.25" align-x="324.75" align-y="1164" 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">y1</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="27"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> . См. рис.2.</inlineAttr> </f> </text> </region> <region region-id="157" left="72" top="1206" width="254.25" height="157.5" align-x="72" align-y="1206" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="28"/> <rendering item-idref="29"/> </region> <region region-id="259" left="54" top="1416.75" width="279.75" height="13.5" align-x="54" align-y="1428" 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">Представим отдельно поверхность Y на рис.2а, 2б.</f> </text> </region> <region region-id="270" left="54" top="1446.75" width="92.25" height="22.5" align-x="98.25" align-y="1458" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y1</ml:id> <ml:sequence> <ml:id xml:space="preserve">n</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="271" left="288" top="1446.75" width="167.25" height="17.25" align-x="330" align-y="1458" 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">Y2</ml:id> <ml:sequence> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">if</ml:id> <ml:sequence> <ml:apply> <ml:notEqual/> <ml:id xml:space="preserve">m</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:real>-1</ml:real> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="31"/> </region> <region region-id="264" left="36" top="1512" width="181.5" height="186.75" align-x="126" align-y="1512" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="32" disable-calc="false"> <inputs> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">Y1</ml:id> </inputs> <outputs/> </component> <rendering item-idref="33"/> </region> <region region-id="265" left="294" top="1512" width="181.5" height="188.25" align-x="384" align-y="1512" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="34" disable-calc="false"> <inputs> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">Y2</ml:id> </inputs> <outputs/> </component> <rendering item-idref="35"/> </region> <region region-id="126" left="60" top="1746.75" width="440.25" height="13.5" align-x="60" align-y="1758" 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 1 </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr underline="false">.<sp count="3"/>Найти решение дифференциального уранения методом Эйлера:</inlineAttr> </f> </text> </region> <region region-id="127" left="192" top="1767" width="101.25" height="36.75" align-x="233.25" align-y="1788" 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:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="36"/> </region> <region region-id="128" left="66" top="1809" width="377.25" height="35.25" align-x="66" align-y="1830" 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">Дано: </f> <region region-id="129" left="101.25" top="1818" width="53.25" height="16.5" align-x="136.5" align-y="1830" 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:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </math> <rendering item-idref="37"/> </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="130" left="162.75" top="1818" width="53.25" height="16.5" align-x="198" align-y="1830" 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:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">y</ml:id> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </math> <rendering item-idref="38"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Начальные условия: </inlineAttr> </f> <region region-id="131" left="339.75" top="1818.75" width="45" height="17.25" align-x="358.5" align-y="1830" 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">x</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>1</ml:real> <ml:real>1</ml:real> </ml:apply> </ml:define> </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">, </inlineAttr> </f> <region region-id="132" left="393" top="1809.75" width="50.25" height="34.5" align-x="411.75" align-y="1830" 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:real>0</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:define> </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">.</inlineAttr> </f> </text> </region> <region region-id="133" left="66" top="1854.75" width="150.75" height="13.5" align-x="66" align-y="1866" 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">Число точек дискретизации:</f> </text> </region> <region region-id="134" left="222" top="1854.75" width="99" height="15.75" align-x="245.25" align-y="1866" 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">N3</ml:id> <ml:apply> <ml:plus/> <ml:real>30</ml:real> <ml:id xml:space="preserve">FRAME</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="135" left="360" top="1854.75" width="70.5" height="15.75" align-x="370.5" align-y="1866" 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">i</ml:id> <ml:range> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">N3</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="42"/> </region> <region region-id="243" left="66" top="1884.75" width="97.5" height="13.5" align-x="66" align-y="1896" 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">Задание функции:</f> </text> </region> <region region-id="242" left="174" top="1884.75" width="84" height="15.75" align-x="211.5" align-y="1896" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="43"/> </region> <region region-id="241" left="282" top="1879.5" width="20.25" height="21" align-x="294" align-y="1896" 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:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">y</ml:id> </ml:apply> </math> <rendering item-idref="44"/> </region> <region region-id="244" left="354" top="1884.75" width="36.75" height="15.75" align-x="366" align-y="1896" 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">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="45"/> </region> <region region-id="138" left="66" top="1914.75" width="250.5" height="13.5" align-x="66" align-y="1926" 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">Область определения функции:<sp count="2"/>x=[x0</f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12">b]</f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> </text> </region> <region region-id="164" left="336" top="1914.75" width="30.75" height="15.75" align-x="350.25" align-y="1926" 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">b</ml:id> <ml:real>3</ml:real> </ml:define> </math> <rendering item-idref="46"/> </region> <region region-id="143" left="66" top="1944.75" width="368.25" height="13.5" align-x="66" align-y="1956" 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">Просмотр на экране дисплея численного значения функции в точке:</f> </text> </region> <region region-id="163" left="72" top="1974.75" width="70.5" height="17.25" align-x="120.75" 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:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="47"/> </region> <region region-id="145" left="66" top="2010.75" width="80.25" height="13.5" align-x="66" align-y="2022" 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">Дискрет по "х":</f> </text> </region> <region region-id="146" left="162" top="2001" width="56.25" height="35.25" align-x="175.5" align-y="2022" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" 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">N3</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="48"/> </region> <region region-id="147" left="66" top="2040.75" width="244.5" height="13.5" align-x="66" align-y="2052" 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">Для выше выбранных параметров он равен:</f> </text> </region> <region region-id="148" left="330" top="2040.75" width="55.5" height="15.75" align-x="342.75" align-y="2052" 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="95589038-740e-4125-9372-18f2106d566d" version-id="7c1fdf6f-791c-4ee9-9b88-3c423b9a2ac4" branch-id="00000000-0000-0000-0000-000000000000" revision-num="10" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\изд-во\ЧислМетРешЗадДиффузии\МетодыЭйАдТейСравАнМет.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="95589038-740e-4125-9372-18f2106d566d" version-id="7c1fdf6f-791c-4ee9-9b88-3c423b9a2ac4" branch-id="00000000-0000-0000-0000-000000000000" revision-num="10" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\изд-во\ЧислМетРешЗадДиффузии\МетодыЭйАдТейСравАнМет.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">e7bc9de7c6f0c5330ae91971ef8d3b69</contentHash> <ml:id xml:space="preserve">h</ml:id> </ml:provenance> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.066666666666666666</ml:real> </result> </ml:eval> </math> <rendering item-idref="49"> <element-image-map> <box left="1.5" top="0.75" width="4.5" height="14.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="149" left="66" top="2064.75" width="221.25" height="13.5" align-x="66" align-y="2076" 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">Дискретное представление переменной:</f> </text> </region> <region region-id="150" left="342" top="2064.75" width="61.5" height="17.25" align-x="357.75" align-y="2076" 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">x</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="50"/> </region> <region region-id="151" left="66" top="2118.75" width="411.75" height="13.5" align-x="66" align-y="2130" 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">Таким образом, для вычисления У(х) методом Эйлера получим выражение:</f> </text> </region> <region region-id="152" left="120" top="2142.75" width="102.75" height="17.25" align-x="147" align-y="2154" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="95589038-740e-4125-9372-18f2106d566d" version-id="7c1fdf6f-791c-4ee9-9b88-3c423b9a2ac4" branch-id="00000000-0000-0000-0000-000000000000" revision-num="10" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\изд-во\ЧислМетРешЗадДиффузии\МетодыЭйАдТейСравАнМет.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="95589038-740e-4125-9372-18f2106d566d" version-id="7c1fdf6f-791c-4ee9-9b88-3c423b9a2ac4" branch-id="00000000-0000-0000-0000-000000000000" revision-num="10" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\изд-во\ЧислМетРешЗадДиффузии\МетодыЭйАдТейСравАнМет.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">db8694df57292068c283f1c423d59777</contentHash> <ml:define> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">i</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:define> </ml:provenance> </math> <rendering item-idref="51"> <element-image-map> <box left="1.5" top="0.75" width="101.25" height="15.75" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="153" left="492" top="2148.75" width="15.75" height="15.75" align-x="498.75" align-y="2160" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="95589038-740e-4125-9372-18f2106d566d" version-id="7c1fdf6f-791c-4ee9-9b88-3c423b9a2ac4" branch-id="00000000-0000-0000-0000-000000000000" revision-num="10" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\изд-во\ЧислМетРешЗадДиффузии\МетодыЭйАдТейСравАнМет.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="95589038-740e-4125-9372-18f2106d566d" version-id="7c1fdf6f-791c-4ee9-9b88-3c423b9a2ac4" branch-id="00000000-0000-0000-0000-000000000000" revision-num="10" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\изд-во\ЧислМетРешЗадДиффузии\МетодыЭйАдТейСравАнМет.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">86c8e9dc912356ce83299d8051318ecf</contentHash> <ml:parens> <ml:real>1</ml:real> </ml:parens> </ml:provenance> </math> <rendering item-idref="52"> <element-image-map> <box left="1.5" top="0.75" width="14.25" height="14.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="168" left="78" top="2178" width="237.75" height="157.5" align-x="78" align-y="2178" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="53"/> <rendering item-idref="54"/> </region> <region region-id="171" left="60" top="2364.75" width="418.5" height="67.5" align-x="60" align-y="2376" 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">С другой стороны это решение можно получить более точными численными методами штатного программного обеспечения Mathcad. Например, исполь-</f> <f family="Arial Cyr" charset="204" size="12">зуя функцию Bulstoer(v,x1,x2,N,D). (См. мет.пособие, Титов, Комп.Технол... ),</f> <f family="Arial Cyr" charset="204" size="12">представляющую матрицу решения системы ОДУ методом Булириш-Штера</f> <f family="Arial Cyr" charset="204" size="12">с постоянным шагом,</f> </text> </region> <region region-id="188" left="66" top="2460.75" width="421.5" height="27" align-x="66" align-y="2472" 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: </f> <f family="Arial Cyr" charset="204" size="12">v - вектор начальных условий; х1,х2 - интервал (начало - конец); N - число</f> <f family="Arial Cyr" charset="204" size="12">точек отсчёта;<sp count="2"/>D - вектор правой части системы ОДУ.</f> </text> </region> <region region-id="172" left="60" top="2502.75" width="45.75" height="13.5" align-x="60" align-y="2514" 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">Пример.</f> </text> </region> <region region-id="173" left="0" top="2529" width="6008.25" align-x="8.25" align-y="2538" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="" height="848.25"> <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="173" bottom-lock-id="174" bottom-tag=""> <region region-id="177" left="30" top="2538.75" width="45.75" height="13.5" align-x="30" align-y="2550" 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">Пример.</f> </text> </region> <region region-id="178" left="36" top="2568.75" width="348" height="115.5" align-x="36" align-y="2580" 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">Пусть задана система линейных дифференциальных уравнений</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="19"/> </f> <region region-id="180" left="96" top="2565.75" width="86.25" height="44.25" align-x="167.25" align-y="2592" 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:minus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:id xml:space="preserve">y</ml:id> </ml:lambda> </ml:apply> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> </math> <rendering item-idref="55"/> </region> <f family="Arial Cyr" charset="204" size="12"> <sp count="18"/> </f> <region region-id="71" left="90" top="2625.75" width="132" height="44.25" align-x="180.75" align-y="2652" 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:apply> <ml:plus/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:id xml:space="preserve">y</ml:id> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">y</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:lambda> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="56"/> </region> </text> </region> <region region-id="179" left="42" top="2700.75" width="442.5" height="82.5" align-x="42" align-y="2712" 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 численное решение исходной системы дифференциальных уравнений средствами MathCAD, используя функцию Bulstoer. Для этого введем обозначе-</f> <f family="Arial Cyr" charset="204" size="12">ния: </f> <region region-id="181" left="66" top="2730.75" width="31.5" height="17.25" align-x="78.75" 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:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="57"/> </region> <f family="Arial Cyr" charset="204" size="12">, </f> <region region-id="182" left="108" top="2730.75" width="31.5" height="17.25" align-x="120.75" 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:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="58"/> </region> <f family="Arial Cyr" charset="204" size="12"> и перепишем исходную систему в нормальной форме:</f> <region region-id="78" left="42" top="2739" width="47.25" height="36.75" align-x="70.5" align-y="2760" 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:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="59"/> </region> <f family="Arial Cyr" charset="204" size="12">,<sp count="5"/> </f> <region region-id="183" left="108" top="2739" width="47.25" height="36.75" align-x="136.5" align-y="2760" 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:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="60"/> </region> <f family="Arial Cyr" charset="204" size="12">,<sp count="5"/> </f> <region region-id="80" left="174" top="2739" width="75.75" height="36.75" align-x="202.5" align-y="2760" 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:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="61"/> </region> <f family="Arial Cyr" charset="204" size="12">,<sp count="6"/> </f> <region region-id="81" left="270" top="2739" width="141" height="36.75" align-x="298.5" align-y="2760" 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:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:real>-2</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </math> <rendering item-idref="62"/> </region> <f family="Arial Cyr" charset="204" size="12">,</f> </text> </region> <region region-id="185" left="42" top="2808.75" width="456.75" height="88.5" align-x="42" align-y="2820" 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">Вектор начальных условий: Z=(1,0,1,-1). Будем искать решение данной системы уравнений в интервале [0, 4 </f> <region region-id="90" left="198" top="2820" width="9" height="16.5" align-x="201.75" align-y="2832" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">π</ml:id> </math> <rendering item-idref="63"/> </region> <f family="Arial Cyr" charset="204" size="12"> ] с количеством точек, в которых ищется прибли-</f> <f family="Arial Cyr" charset="204" size="12">женное решение, равным 100, используя функцию Bulstoer(z,0,4<sp count="2"/> </f> <region region-id="89" left="396" top="2838" width="9" height="16.5" align-x="399.75" align-y="2850" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">π</ml:id> </math> <rendering item-idref="64"/> </region> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/>,100,D). Результатом решения будет матрица, содержащая rows(R) строк и cols(R) столб-</f> <f family="Arial Cyr" charset="204" size="12">цов. Затем зададим ранжированную переменную:<sp count="2"/>i=0..rows(R)-1. Ниже повторе-</f> <f family="Arial Cyr" charset="204" size="12">на эта запись в терминах MathCAD:</f> </text> </region> <region region-id="189" left="90" top="2924.25" width="168.75" height="84.75" align-x="129.75" align-y="2970" 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">D</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:matrix rows="4" cols="1"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:real>-2</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:matrix> </ml:define> </math> <rendering item-idref="65"/> </region> <region region-id="201" left="330" top="2927.25" width="57.75" height="78.75" align-x="348" align-y="2970" 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">vz</ml:id> <ml:matrix rows="4" cols="1"> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>-1</ml:real> </ml:matrix> </ml:define> </math> <rendering item-idref="66"/> </region> <region region-id="202" left="438" top="2927.25" width="24.75" height="78.75" align-x="449.25" align-y="2970" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:matrix rows="4" cols="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">x'</ml:id> <ml:id xml:space="preserve">y'</ml:id> </ml:matrix> </math> <rendering item-idref="67"/> </region> <region region-id="196" left="66" top="3036" width="159" height="16.5" align-x="81.75" align-y="3048" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">R</ml:id> <ml:apply> <ml:id xml:space="preserve">Bulstoer</ml:id> <ml:sequence> <ml:id xml:space="preserve">vz</ml:id> <ml:real>0</ml:real> <ml:apply> <ml:mult style="default"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>100</ml:real> <ml:id xml:space="preserve">D</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="68"/> </region> <region region-id="192" left="240" top="3036.75" width="95.25" height="15.75" align-x="250.5" align-y="3048" 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">i</ml:id> <ml:range> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">rows</ml:id> <ml:id xml:space="preserve">R</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="69"/> </region> <region region-id="199" left="66" top="3072.75" width="81" height="15.75" align-x="113.25" align-y="3084" 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">rows</ml:id> <ml:id xml:space="preserve">R</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>101</ml:real> </result> </ml:eval> </math> <rendering item-idref="70"/> </region> <region region-id="200" left="168" top="3072.75" width="64.5" height="15.75" align-x="210.75" align-y="3084" 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">cols</ml:id> <ml:id xml:space="preserve">R</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>5</ml:real> </result> </ml:eval> </math> <rendering item-idref="71"/> </region> <region region-id="204" left="54" top="3108" width="171.75" height="143.25" align-x="54" align-y="3108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="72"/> <rendering item-idref="73"/> </region> <region region-id="205" left="282" top="3156.75" width="98.25" height="13.5" align-x="282" align-y="3168" 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">Фазовый портрет.</f> </text> </region> <region region-id="208" left="42" top="3283.5" width="430.5" height="66" align-x="42" align-y="3300" 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">Здесь равносильны записи: </f> <region region-id="207" left="198" top="3284.25" width="71.25" height="24.75" align-x="239.25" align-y="3300" 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:parens> <ml:apply> <ml:matcol/> <ml:id xml:space="preserve">R</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">R</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:real>2</ml:real> </ml:sequence> </ml:apply> </ml:apply> </math> <rendering item-idref="74"/> </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="209" left="282" top="3284.25" width="71.25" height="24.75" align-x="323.25" align-y="3300" 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:parens> <ml:apply> <ml:matcol/> <ml:id xml:space="preserve">R</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">R</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:real>1</ml:real> </ml:sequence> </ml:apply> </ml:apply> </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">. В дискретном виде в</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">первом столбце матрицы R записано решение x(t), во втором -<sp count="2"/>y(t), третьем и четвёртом - значения их производных<sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">x'(t),<sp count="2"/>y'(t); в нулевом столбце идёт нуме-</f> <f family="Arial Cyr" charset="204" size="12">рация строк<sp count="2"/>или, что одно и тоже, запись<sp count="2"/>t в </f> <f family="Arial Cyr" charset="204" size="12">дискретном виде.</f> </text> </region> </area> <rendering item-idref="76"/> </region> <region region-id="212" left="60" top="3396.75" width="415.5" height="40.5" align-x="60" align-y="3408" 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">Аналогичное назначение имеет функция Rkadapt(</f> <f family="Arial Cyr" charset="204" size="12">v,x1,x2,N,D</f> <f family="Arial Cyr" charset="204" size="12">) с переменным</f> <f family="Arial Cyr" charset="204" size="12">шагом и<sp count="2"/>r</f> <f family="Arial Cyr" charset="204" size="12">kfixed(</f> <f family="Arial Cyr" charset="204" size="12">v,x1,x2,N,D</f> <f family="Arial Cyr" charset="204" size="12">) с постоянным шагом, в которых реализован ме-</f> <f family="Arial Cyr" charset="204" size="12">тод Рунге-Кутта.</f> </text> </region> <region region-id="214" left="60" top="3450.75" width="357.75" height="31.5" align-x="60" align-y="3462" 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">Итак, проведём решение ОДУ упр.1 численно, используя </f> <f family="Arial Cyr" charset="204" size="12">Bulstoer:</f> <f family="Arial Cyr" charset="204" size="12">(вместо</f> <region region-id="236" left="102" top="3462.75" width="11.25" height="17.25" align-x="106.5" align-y="3474" 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">y</ml:id> <ml:real>0</ml:real> </ml:apply> </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"> <sp count="2"/>можно брать<sp count="2"/> </inlineAttr> </f> <region region-id="237" left="198" top="3462.75" width="12" height="15.75" align-x="203.25" align-y="3474" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">y0</ml:id> </math> <rendering item-idref="78"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> и наоборот</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">)</f> </text> </region> <region region-id="218" left="66" top="3510.75" width="102.75" height="17.25" align-x="108.75" align-y="3522" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">D</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> </ml:function> <ml:matrix rows="1" cols="1"> <ml:apply> <ml:id 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<ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">rows</ml:id> <ml:id xml:space="preserve">R1</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="82"/> </region> <region region-id="229" left="72" top="3588.75" width="81" height="15.75" align-x="125.25" align-y="3600" 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">rows</ml:id> <ml:id xml:space="preserve">R1</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>31</ml:real> </result> </ml:eval> </math> <rendering item-idref="83"/> </region> <region region-id="230" left="174" top="3588.75" width="70.5" height="15.75" align-x="222.75" align-y="3600" 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">cols</ml:id> <ml:id xml:space="preserve">R1</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </result> </ml:eval> </math> <rendering item-idref="84"/> </region> <region region-id="234" left="24" top="3624" width="350.25" height="191.25" align-x="24" align-y="3624" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="85"/> <rendering item-idref="86"/> </region> <region region-id="235" left="48" top="3840.75" width="431.25" height="30" align-x="48" 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="false"> <f family="Arial Cyr" charset="204" size="12">Из рис.3а видно, что метод Эйлера менее точен. Однако, с уменьшением диск-</f> <f family="Arial Cyr" charset="204" size="12">рета решение y(x)<sp count="2"/>стремится к R1</f> <f family="Arial Cyr" charset="204" size="12"> i,1 </f> <f family="Arial Cyr" charset="204" size="12">.</f> </text> </region> <region region-id="245" left="60" top="3876.75" width="440.25" height="13.5" align-x="60" align-y="3888" 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 2 </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr underline="false">.<sp count="3"/>Найти решение дифференциального уранения методом Эйлера:</inlineAttr> </f> </text> </region> <region region-id="246" left="192" top="3897" width="69.75" height="36.75" align-x="233.25" align-y="3918" 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:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">y</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="87"/> </region> <region region-id="248" left="48" top="3942.75" width="305.25" height="21.75" align-x="48" align-y="3960" 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">В упр.1 надо f(x,y) заменить на </f> <region region-id="249" left="216" top="3943.5" width="20.25" height="21" align-x="228" align-y="3960" 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:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">y</ml:id> </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"> <sp count="2"/>и получить решение.</inlineAttr> </f> </text> </region> </regions> <binaryContent> <item item-id="1" content-encoding="gzip">H4sIAAAAAAAA/4yNvQ6CQBCEdzlQwd/GGhMbj9InsCBWJj6CnvifEA1SWPrm5+yCvUf2GGa/ GQZExBiHSVQb3OPjcnep3PO6Oz+q0tURyZljYrVzVzu1aIHplcX2cD8VdWPtmxcFM1ym7Z9K q2r5Stz79lprtVE2w2y4ZZvj8RBn/7u/8m5dueK0zANd2RZMeUJs2Ytitlj6lAO2BtGUDdvQ 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Arial;}Arial CYR;}} \viewkind4\uc1\pard\qc\f0\fs18\'d2\'e8\'f2\'ee\'e2\par }</center> <right>{\rtf1\ansi\ansicpg1251\deff0\deflang1049{\fonttbl{\f0\fnil\fprq15\fcharset204{\*\fname Arial;}Arial CYR;}} \viewkind4\uc1\pard\qr\f0\fs18\{n\}\par }</right> </footer> </pageModel> <colorModel background-color="#fff" default-highlight-color="#ffff80"/> <language math="EN" UI="EN"/> </presentation> <calculation> <builtInVariables array-origin="0" convergence-tolerance="0.001" constraint-tolerance="0.001" random-seed="1" prn-precision="4" prn-col-width="8"/> <calculationBehavior automatic-recalculation="true" matrix-strict-singularity-check="true" optimize-expressions="false" exact-boolean="true" strings-use-origin="false" zero-over-zero="0"> <compatibility multiple-assignment="MC12" local-assignment="MC12"/> </calculationBehavior> <units> <currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor protection-level="none" view-annotations="false" view-regions="false"> <ruler is-visible="false" 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="277" 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="1" left="42" top="13.5" width="407.25" height="51" align-x="42" 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="true"> <f family="Arial Cyr" charset="204" size="12"> Геометрическая интерпретация решения дифференциального уравнения </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <region region-id="2" left="103.5" top="27.75" width="84" height="36.75" align-x="147" align-y="48.75" 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:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:sequence> </ml:apply> </ml:apply> </math> <rendering item-idref="2"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">.</inlineAttr> </f> </text> </region> <region region-id="4" left="48" top="72" width="442.5" height="90" align-x="48" align-y="84" 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">Рассмотрим на примере уравнения с </f> <region region-id="5" left="260.25" top="72.75" width="78.75" height="15.75" align-x="297.75" align-y="84" show-border="false" show-highlight="true" 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">f</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="3"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Решением этого диф-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">ференциального уравнения будет семейство кривых </inlineAttr> </f> <region region-id="6" left="348.75" top="89.25" width="93.75" height="15.75" align-x="383.25" align-y="100.5" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math error="bad_dimensions Unitless Charge " optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">y1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:id xml:space="preserve">C</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="4"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, где</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">константа С определяется начальными условиями: x</f> </inlineAttr> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12"> o </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, y(x</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> o </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">)=y</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> o </inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">. </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">Функция f(x,y) задает поверхность, а y1(x), кривая под этой поверхностью, явля-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">ется решением </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">дифференциального уравнения, в каждой точке которого угловой коэффициент касательной равен значению f(x,y) в этой точке.</f> </text> </region> <region region-id="12" left="48" top="174.75" width="441.75" height="87.75" align-x="48" align-y="186" 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">Итак, зададим поверхность f(x,y) в виде дискретных точек и запишем её в</f> <f family="Arial Cyr" charset="204" size="12">матрицу М. Пусть </f> <region region-id="8" left="150" top="189.75" width="62.25" height="16.5" align-x="194.25" align-y="201.75" 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:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </math> <rendering item-idref="5"/> </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="9" left="216" top="190.5" width="57.75" height="15.75" align-x="258" align-y="201.75" 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:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:real>-2</ml:real> <ml:id xml:space="preserve">y</ml:id> </ml:apply> <ml:real>2</ml:real> </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">. В качестве дискретного шага по</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">переменным x и у выберем </inlineAttr> </f> <region region-id="13" left="205.5" top="207" width="45" height="15.75" align-x="219" align-y="218.25" show-border="false" show-highlight="true" 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:real>0.04</ml:real> </ml:define> </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"> (здесь он один и тот же, но может быть</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">разным и переменным)</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Введём обозначения: n - </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">номер строки матрицы </f> <f family="Arial Cyr" charset="204" size="12">М; </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">m - номер столбца матрицы М. Число строк и столбцов матрицы М, </f> </inlineAttr> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">определим как N:</f> </inlineAttr> </text> </region> <region region-id="52" left="60" top="273.75" width="90" height="34.5" align-x="83.25" align-y="294" 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">N1</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="76" left="180" top="282.75" width="56.25" height="15.75" align-x="202.5" align-y="294" 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">N1</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>157</ml:real> </result> </ml:eval> </math> <rendering item-idref="9"/> </region> <region region-id="75" left="252" top="282.75" width="54.75" height="15.75" align-x="265.5" align-y="294" 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">n</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">N1</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="277" left="60" top="322.5" width="78" height="33.75" align-x="83.25" align-y="342" 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">N2</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="11"/> </region> <region region-id="276" left="186" top="330.75" width="50.25" height="15.75" align-x="208.5" align-y="342" 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">N2</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>50</ml:real> </result> </ml:eval> </math> <rendering item-idref="12"/> </region> <region region-id="275" left="276" top="330.75" width="57.75" height="15.75" align-x="292.5" align-y="342" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">m</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">N2</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="26" left="54" top="366.75" width="210" height="13.5" align-x="54" align-y="378" 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">Перейдем к дискретным переменным:</f> </text> </region> <region region-id="33" left="288" top="366" width="72" height="18" align-x="306.75" align-y="378" 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">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="97" left="390" top="366.75" width="75.75" height="17.25" align-x="411.75" align-y="378" 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:id xml:space="preserve">m</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:real>-1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="62" left="60" top="390.75" width="67.5" height="17.25" align-x="78" align-y="402" 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>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-3.1415926535897931</ml:real> </result> </ml:eval> </math> <rendering item-idref="16"/> </region> <region region-id="63" left="144" top="390.75" width="72.75" height="17.25" align-x="167.25" align-y="402" 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>78</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.021592653589793009</ml:real> </result> </ml:eval> </math> <rendering item-idref="17"/> </region> <region region-id="64" left="264" top="390.75" width="66" height="17.25" align-x="287.25" align-y="402" 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>79</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.018407346410207026</ml:real> </result> </ml:eval> </math> <rendering item-idref="18"/> </region> <region region-id="39" left="54" top="420" width="447.75" height="60" align-x="54" align-y="432" 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:<sp count="2"/> </f> <region region-id="41" left="96" top="420.75" width="93" height="17.25" align-x="134.25" align-y="432" 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">M</ml:id> <ml:sequence> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="19"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Так как f(x,y) не зависит от </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" underline="false" line-through="false">у</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, то у матрицы </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" underline="false" line-through="false"> М </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> все </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">столбцы будут одинаковы и задавать она будет цилиндрическую поверхность с образующей, параллельной оси ординат (0</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" underline="false" line-through="false">у</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">), а направляющая будет лежать в плоскости </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" underline="false" line-through="false">f</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">0</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="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">(аппликата-абсцисса)</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Теперь построим поверхность f(x,y) рис.1.</inlineAttr> </f> </text> </region> <region region-id="79" left="54" top="492" width="450.75" height="94.5" align-x="54" align-y="504" 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 матрицу решений </f> <f family="Arial Cyr" charset="204" size="12"> y </f> <f family="Arial Cyr" charset="204" size="12">1(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">)=sin(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">)+</f> <f family="Arial Cyr" charset="204" size="12"> C </f> <f family="Arial Cyr" charset="204" size="12">, обозначив её через </f> <f family="Arial Cyr" charset="204" size="12"> Y </f> <f family="Arial Cyr" charset="204" size="12">. </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">Пусть </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">С</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">=1. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">Зависимость </inlineAttr> </f> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12"> y </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">1(</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">x</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">) представим в дискретном виде и запишем в матрицу </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">Y</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">. Причем номер строки </inlineAttr> </f> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12"> n </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal"> матрицы будет абсциссой, а номер столбца </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">m</inlineAttr> </f> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12">=v</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">n</inlineAttr> </f> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12"> -<sp count="2"/>ординатой решения </f> </inlineAttr> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12">y</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">1(</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">x</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">). Сам же элемент матрицы </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">Y</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal"> n,m </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal"> будет представлять аппликату образующей цилиндрической поверхности с направляющей </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">y</inlineAttr> </f> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12">1(</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">x</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">).<sp count="2"/> </inlineAttr> </f> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12">Таким образом, зависимость </f> </inlineAttr> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12"> y </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">1(</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">x</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">) можно представить </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">вектором </inlineAttr> </f> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12"> v </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal"> n </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">.</inlineAttr> </f> </text> </region> <region region-id="121" left="60" top="604.5" width="75.75" height="33.75" align-x="82.5" align-y="624" 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">C1</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">FRAME</ml:id> <ml:real>10</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="122" left="180" top="603" width="148.5" height="35.25" align-x="198.75" align-y="624" 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">v</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">C1</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="123" left="360" top="612.75" width="45.75" height="17.25" align-x="378" align-y="624" 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">v</ml:id> <ml:real>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>24</ml:real> </result> </ml:eval> </math> <rendering item-idref="22"/> </region> <region region-id="124" left="438" top="612.75" width="48" height="17.25" align-x="464.25" align-y="624" 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">v</ml:id> <ml:id xml:space="preserve">N2</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </result> </ml:eval> </math> <rendering item-idref="23"/> </region> <region region-id="112" left="138" top="660.75" width="86.25" height="22.5" align-x="176.25" align-y="672" 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:id xml:space="preserve">n</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="77" left="60" top="750" width="429.75" height="326.25" align-x="273.75" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="25" disable-calc="false"> <inputs> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">M</ml:id> <ml:id xml:space="preserve">Y</ml:id> </ml:sequence> </inputs> <outputs/> </component> <rendering item-idref="26"/> </region> <region region-id="125" left="60" top="1098.75" width="458.25" height="40.5" align-x="60" align-y="1110" 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">Из рис.1 видно, что </f> <f family="Arial Cyr" charset="204" size="12">в каждой точке y1(x) угловой коэффициент касательной </f> <f family="Arial Cyr" charset="204" size="12">равен значению f(x,y) в этой точке. А поверхность f(x,y) является геометри-</f> <f family="Arial Cyr" charset="204" size="12">ческим местом таких точек. </f> </text> </region> <region region-id="158" left="66" top="1152" width="372" height="22.5" align-x="66" align-y="1164" 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">Убедимся в том, что v</f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12">h</f> <f family="Arial Cyr" charset="204" size="12">-1 есть синусоида: </f> <region region-id="159" left="300" top="1152.75" width="73.5" height="17.25" align-x="324.75" align-y="1164" 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">y1</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="27"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> . См. рис.2.</inlineAttr> </f> </text> </region> <region region-id="157" left="72" top="1206" width="254.25" height="157.5" align-x="72" align-y="1206" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="28"/> <rendering item-idref="29"/> </region> <region region-id="259" left="54" top="1416.75" width="279.75" height="13.5" align-x="54" align-y="1428" 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">Представим отдельно поверхность Y на рис.2а, 2б.</f> </text> </region> <region region-id="270" left="54" top="1446.75" width="92.25" height="22.5" align-x="98.25" align-y="1458" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Y1</ml:id> <ml:sequence> <ml:id xml:space="preserve">n</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="271" left="288" top="1446.75" width="167.25" height="17.25" align-x="330" align-y="1458" 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">Y2</ml:id> <ml:sequence> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">if</ml:id> <ml:sequence> <ml:apply> <ml:notEqual/> <ml:id xml:space="preserve">m</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:real>-1</ml:real> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="31"/> </region> <region region-id="264" left="36" top="1512" width="181.5" height="186.75" align-x="126" align-y="1512" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="32" disable-calc="false"> <inputs> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">Y1</ml:id> </inputs> <outputs/> </component> <rendering item-idref="33"/> </region> <region region-id="265" left="294" top="1512" width="181.5" height="188.25" align-x="384" align-y="1512" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="34" disable-calc="false"> <inputs> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">Y2</ml:id> </inputs> <outputs/> </component> <rendering item-idref="35"/> </region> <region region-id="126" left="60" top="1746.75" width="440.25" height="13.5" align-x="60" align-y="1758" 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 1 </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr underline="false">.<sp count="3"/>Найти решение дифференциального уранения методом Эйлера:</inlineAttr> </f> </text> </region> <region region-id="127" left="192" top="1767" width="101.25" height="36.75" align-x="233.25" align-y="1788" 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:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="36"/> </region> <region region-id="128" left="66" top="1809" width="377.25" height="35.25" align-x="66" align-y="1830" 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">Дано: </f> <region region-id="129" left="101.25" top="1818" width="53.25" height="16.5" align-x="136.5" align-y="1830" 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:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </math> <rendering item-idref="37"/> </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="130" left="162.75" top="1818" width="53.25" height="16.5" align-x="198" align-y="1830" 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:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">y</ml:id> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </math> <rendering item-idref="38"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Начальные условия: </inlineAttr> </f> <region region-id="131" left="339.75" top="1818.75" width="45" height="17.25" align-x="358.5" align-y="1830" 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">x</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>1</ml:real> <ml:real>1</ml:real> </ml:apply> </ml:define> </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">, </inlineAttr> </f> <region region-id="132" left="393" top="1809.75" width="50.25" height="34.5" align-x="411.75" align-y="1830" 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:real>0</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:define> </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">.</inlineAttr> </f> </text> </region> <region region-id="133" left="66" top="1854.75" width="150.75" height="13.5" align-x="66" align-y="1866" 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">Число точек дискретизации:</f> </text> </region> <region region-id="134" left="222" top="1854.75" width="99" height="15.75" align-x="245.25" align-y="1866" 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">N3</ml:id> <ml:apply> <ml:plus/> <ml:real>30</ml:real> <ml:id xml:space="preserve">FRAME</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="135" left="360" top="1854.75" width="70.5" height="15.75" align-x="370.5" align-y="1866" 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">i</ml:id> <ml:range> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">N3</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="42"/> </region> <region region-id="243" left="66" top="1884.75" width="97.5" height="13.5" align-x="66" align-y="1896" 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">Задание функции:</f> </text> </region> <region region-id="242" left="174" top="1884.75" width="84" height="15.75" align-x="211.5" align-y="1896" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="43"/> </region> <region region-id="241" left="282" top="1879.5" width="20.25" height="21" align-x="294" align-y="1896" 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:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">y</ml:id> </ml:apply> </math> <rendering item-idref="44"/> </region> <region region-id="244" left="354" top="1884.75" width="36.75" height="15.75" align-x="366" align-y="1896" 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">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="45"/> </region> <region region-id="138" left="66" top="1914.75" width="250.5" height="13.5" align-x="66" align-y="1926" 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">Область определения функции:<sp count="2"/>x=[x0</f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12">b]</f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> </text> </region> <region region-id="164" left="336" top="1914.75" width="30.75" height="15.75" align-x="350.25" align-y="1926" 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">b</ml:id> <ml:real>3</ml:real> </ml:define> </math> <rendering item-idref="46"/> </region> <region region-id="143" left="66" top="1944.75" width="368.25" height="13.5" align-x="66" align-y="1956" 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">Просмотр на экране дисплея численного значения функции в точке:</f> </text> </region> <region region-id="163" left="72" top="1974.75" width="70.5" height="17.25" align-x="120.75" 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:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="47"/> </region> <region region-id="145" left="66" top="2010.75" width="80.25" height="13.5" align-x="66" align-y="2022" 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">Дискрет по "х":</f> </text> </region> <region region-id="146" left="162" top="2001" width="56.25" height="35.25" align-x="175.5" align-y="2022" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" 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">N3</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="48"/> </region> <region region-id="147" left="66" top="2040.75" width="244.5" height="13.5" align-x="66" align-y="2052" 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">Для выше выбранных параметров он равен:</f> </text> </region> <region region-id="148" left="330" top="2040.75" width="55.5" height="15.75" align-x="342.75" align-y="2052" 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="95589038-740e-4125-9372-18f2106d566d" version-id="7c1fdf6f-791c-4ee9-9b88-3c423b9a2ac4" branch-id="00000000-0000-0000-0000-000000000000" revision-num="10" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\изд-во\ЧислМетРешЗадДиффузии\МетодыЭйАдТейСравАнМет.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="95589038-740e-4125-9372-18f2106d566d" version-id="7c1fdf6f-791c-4ee9-9b88-3c423b9a2ac4" branch-id="00000000-0000-0000-0000-000000000000" revision-num="10" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\изд-во\ЧислМетРешЗадДиффузии\МетодыЭйАдТейСравАнМет.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">e7bc9de7c6f0c5330ae91971ef8d3b69</contentHash> <ml:id xml:space="preserve">h</ml:id> </ml:provenance> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.066666666666666666</ml:real> </result> </ml:eval> </math> <rendering item-idref="49"> <element-image-map> <box left="1.5" top="0.75" width="4.5" height="14.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="149" left="66" top="2064.75" width="221.25" height="13.5" align-x="66" align-y="2076" 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">Дискретное представление переменной:</f> </text> </region> <region region-id="150" left="342" top="2064.75" width="61.5" height="17.25" align-x="357.75" align-y="2076" 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">x</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="50"/> </region> <region region-id="151" left="66" top="2118.75" width="411.75" height="13.5" align-x="66" align-y="2130" 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">Таким образом, для вычисления У(х) методом Эйлера получим выражение:</f> </text> </region> <region region-id="152" left="120" top="2142.75" width="102.75" height="17.25" align-x="147" align-y="2154" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="95589038-740e-4125-9372-18f2106d566d" version-id="7c1fdf6f-791c-4ee9-9b88-3c423b9a2ac4" branch-id="00000000-0000-0000-0000-000000000000" revision-num="10" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\изд-во\ЧислМетРешЗадДиффузии\МетодыЭйАдТейСравАнМет.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="95589038-740e-4125-9372-18f2106d566d" version-id="7c1fdf6f-791c-4ee9-9b88-3c423b9a2ac4" branch-id="00000000-0000-0000-0000-000000000000" revision-num="10" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\изд-во\ЧислМетРешЗадДиффузии\МетодыЭйАдТейСравАнМет.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">db8694df57292068c283f1c423d59777</contentHash> <ml:define> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">i</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:define> </ml:provenance> </math> <rendering item-idref="51"> <element-image-map> <box left="1.5" top="0.75" width="101.25" height="15.75" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="153" left="492" top="2148.75" width="15.75" height="15.75" align-x="498.75" align-y="2160" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="95589038-740e-4125-9372-18f2106d566d" version-id="7c1fdf6f-791c-4ee9-9b88-3c423b9a2ac4" branch-id="00000000-0000-0000-0000-000000000000" revision-num="10" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\изд-во\ЧислМетРешЗадДиффузии\МетодыЭйАдТейСравАнМет.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="95589038-740e-4125-9372-18f2106d566d" version-id="7c1fdf6f-791c-4ee9-9b88-3c423b9a2ac4" branch-id="00000000-0000-0000-0000-000000000000" revision-num="10" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\изд-во\ЧислМетРешЗадДиффузии\МетодыЭйАдТейСравАнМет.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">86c8e9dc912356ce83299d8051318ecf</contentHash> <ml:parens> <ml:real>1</ml:real> </ml:parens> </ml:provenance> </math> <rendering item-idref="52"> <element-image-map> <box left="1.5" top="0.75" width="14.25" height="14.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="168" left="78" top="2178" width="237.75" height="157.5" align-x="78" align-y="2178" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="53"/> <rendering item-idref="54"/> </region> <region region-id="171" left="60" top="2364.75" width="418.5" height="67.5" align-x="60" align-y="2376" 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">С другой стороны это решение можно получить более точными численными методами штатного программного обеспечения Mathcad. Например, исполь-</f> <f family="Arial Cyr" charset="204" size="12">зуя функцию Bulstoer(v,x1,x2,N,D). (См. мет.пособие, Титов, Комп.Технол... ),</f> <f family="Arial Cyr" charset="204" size="12">представляющую матрицу решения системы ОДУ методом Булириш-Штера</f> <f family="Arial Cyr" charset="204" size="12">с постоянным шагом,</f> </text> </region> <region region-id="188" left="66" top="2460.75" width="421.5" height="27" align-x="66" align-y="2472" 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: </f> <f family="Arial Cyr" charset="204" size="12">v - вектор начальных условий; х1,х2 - интервал (начало - конец); N - число</f> <f family="Arial Cyr" charset="204" size="12">точек отсчёта;<sp count="2"/>D - вектор правой части системы ОДУ.</f> </text> </region> <region region-id="172" left="60" top="2502.75" width="45.75" height="13.5" align-x="60" align-y="2514" 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">Пример.</f> </text> </region> <region region-id="173" left="0" top="2529" width="6008.25" align-x="8.25" align-y="2538" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="" height="848.25"> <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="173" bottom-lock-id="174" bottom-tag=""> <region region-id="177" left="30" top="2538.75" width="45.75" height="13.5" align-x="30" align-y="2550" 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">Пример.</f> </text> </region> <region region-id="178" left="36" top="2568.75" width="348" height="115.5" align-x="36" align-y="2580" 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">Пусть задана система линейных дифференциальных уравнений</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="19"/> </f> <region region-id="180" left="96" top="2565.75" width="86.25" height="44.25" align-x="167.25" align-y="2592" 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:minus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:id xml:space="preserve">y</ml:id> </ml:lambda> </ml:apply> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> </math> <rendering item-idref="55"/> </region> <f family="Arial Cyr" charset="204" size="12"> <sp count="18"/> </f> <region region-id="71" left="90" top="2625.75" width="132" height="44.25" align-x="180.75" align-y="2652" 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:apply> <ml:plus/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:id xml:space="preserve">y</ml:id> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">y</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:lambda> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="56"/> </region> </text> </region> <region region-id="179" left="42" top="2700.75" width="442.5" height="82.5" align-x="42" align-y="2712" 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 численное решение исходной системы дифференциальных уравнений средствами MathCAD, используя функцию Bulstoer. Для этого введем обозначе-</f> <f family="Arial Cyr" charset="204" size="12">ния: </f> <region region-id="181" left="66" top="2730.75" width="31.5" height="17.25" align-x="78.75" 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:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="57"/> </region> <f family="Arial Cyr" charset="204" size="12">, </f> <region region-id="182" left="108" top="2730.75" width="31.5" height="17.25" align-x="120.75" 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:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="58"/> </region> <f family="Arial Cyr" charset="204" size="12"> и перепишем исходную систему в нормальной форме:</f> <region region-id="78" left="42" top="2739" width="47.25" height="36.75" align-x="70.5" align-y="2760" 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:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="59"/> </region> <f family="Arial Cyr" charset="204" size="12">,<sp count="5"/> </f> <region region-id="183" left="108" top="2739" width="47.25" height="36.75" align-x="136.5" align-y="2760" 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:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="60"/> </region> <f family="Arial Cyr" charset="204" size="12">,<sp count="5"/> </f> <region region-id="80" left="174" top="2739" width="75.75" height="36.75" align-x="202.5" align-y="2760" 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:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="61"/> </region> <f family="Arial Cyr" charset="204" size="12">,<sp count="6"/> </f> <region region-id="81" left="270" top="2739" width="141" height="36.75" align-x="298.5" align-y="2760" 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:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:real>-2</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </math> <rendering item-idref="62"/> </region> <f family="Arial Cyr" charset="204" size="12">,</f> </text> </region> <region region-id="185" left="42" top="2808.75" width="456.75" height="88.5" align-x="42" align-y="2820" 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">Вектор начальных условий: Z=(1,0,1,-1). Будем искать решение данной системы уравнений в интервале [0, 4 </f> <region region-id="90" left="198" top="2820" width="9" height="16.5" align-x="201.75" align-y="2832" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">π</ml:id> </math> <rendering item-idref="63"/> </region> <f family="Arial Cyr" charset="204" size="12"> ] с количеством точек, в которых ищется прибли-</f> <f family="Arial Cyr" charset="204" size="12">женное решение, равным 100, используя функцию Bulstoer(z,0,4<sp count="2"/> </f> <region region-id="89" left="396" top="2838" width="9" height="16.5" align-x="399.75" align-y="2850" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">π</ml:id> </math> <rendering item-idref="64"/> </region> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/>,100,D). Результатом решения будет матрица, содержащая rows(R) строк и cols(R) столб-</f> <f family="Arial Cyr" charset="204" size="12">цов. Затем зададим ранжированную переменную:<sp count="2"/>i=0..rows(R)-1. Ниже повторе-</f> <f family="Arial Cyr" charset="204" size="12">на эта запись в терминах MathCAD:</f> </text> </region> <region region-id="189" left="90" top="2924.25" width="168.75" height="84.75" align-x="129.75" align-y="2970" 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">D</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:matrix rows="4" cols="1"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:real>-2</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:matrix> </ml:define> </math> <rendering item-idref="65"/> </region> <region region-id="201" left="330" top="2927.25" width="57.75" height="78.75" align-x="348" align-y="2970" 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">vz</ml:id> <ml:matrix rows="4" cols="1"> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>-1</ml:real> </ml:matrix> </ml:define> </math> <rendering item-idref="66"/> </region> <region region-id="202" left="438" top="2927.25" width="24.75" height="78.75" align-x="449.25" align-y="2970" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:matrix rows="4" cols="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">x'</ml:id> <ml:id xml:space="preserve">y'</ml:id> </ml:matrix> </math> <rendering item-idref="67"/> </region> <region region-id="196" left="66" top="3036" width="159" height="16.5" align-x="81.75" align-y="3048" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">R</ml:id> <ml:apply> <ml:id xml:space="preserve">Bulstoer</ml:id> <ml:sequence> <ml:id xml:space="preserve">vz</ml:id> <ml:real>0</ml:real> <ml:apply> <ml:mult style="default"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>100</ml:real> <ml:id xml:space="preserve">D</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="68"/> </region> <region region-id="192" left="240" top="3036.75" width="95.25" height="15.75" align-x="250.5" align-y="3048" 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">i</ml:id> <ml:range> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">rows</ml:id> <ml:id xml:space="preserve">R</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="69"/> </region> <region region-id="199" left="66" top="3072.75" width="81" height="15.75" align-x="113.25" align-y="3084" 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">rows</ml:id> <ml:id xml:space="preserve">R</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>101</ml:real> </result> </ml:eval> </math> <rendering item-idref="70"/> </region> <region region-id="200" left="168" top="3072.75" width="64.5" height="15.75" align-x="210.75" align-y="3084" 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">cols</ml:id> <ml:id xml:space="preserve">R</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>5</ml:real> </result> </ml:eval> </math> <rendering item-idref="71"/> </region> <region region-id="204" left="54" top="3108" width="171.75" height="143.25" align-x="54" align-y="3108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="72"/> <rendering item-idref="73"/> </region> <region region-id="205" left="282" top="3156.75" width="98.25" height="13.5" align-x="282" align-y="3168" 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">Фазовый портрет.</f> </text> </region> <region region-id="208" left="42" top="3283.5" width="430.5" height="66" align-x="42" align-y="3300" 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">Здесь равносильны записи: </f> <region region-id="207" left="198" top="3284.25" width="71.25" height="24.75" align-x="239.25" align-y="3300" 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:parens> <ml:apply> <ml:matcol/> <ml:id xml:space="preserve">R</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">R</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:real>2</ml:real> </ml:sequence> </ml:apply> </ml:apply> </math> <rendering item-idref="74"/> </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="209" left="282" top="3284.25" width="71.25" height="24.75" align-x="323.25" align-y="3300" 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:parens> <ml:apply> <ml:matcol/> <ml:id xml:space="preserve">R</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">R</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:real>1</ml:real> </ml:sequence> </ml:apply> </ml:apply> </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">. В дискретном виде в</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">первом столбце матрицы R записано решение x(t), во втором -<sp count="2"/>y(t), третьем и четвёртом - значения их производных<sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">x'(t),<sp count="2"/>y'(t); в нулевом столбце идёт нуме-</f> <f family="Arial Cyr" charset="204" size="12">рация строк<sp count="2"/>или, что одно и тоже, запись<sp count="2"/>t в </f> <f family="Arial Cyr" charset="204" size="12">дискретном виде.</f> </text> </region> </area> <rendering item-idref="76"/> </region> <region region-id="212" left="60" top="3396.75" width="415.5" height="40.5" align-x="60" align-y="3408" 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">Аналогичное назначение имеет функция Rkadapt(</f> <f family="Arial Cyr" charset="204" size="12">v,x1,x2,N,D</f> <f family="Arial Cyr" charset="204" size="12">) с переменным</f> <f family="Arial Cyr" charset="204" size="12">шагом и<sp count="2"/>r</f> <f family="Arial Cyr" charset="204" size="12">kfixed(</f> <f family="Arial Cyr" charset="204" size="12">v,x1,x2,N,D</f> <f family="Arial Cyr" charset="204" size="12">) с постоянным шагом, в которых реализован ме-</f> <f family="Arial Cyr" charset="204" size="12">тод Рунге-Кутта.</f> </text> </region> <region region-id="214" left="60" top="3450.75" width="357.75" height="31.5" align-x="60" align-y="3462" 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">Итак, проведём решение ОДУ упр.1 численно, используя </f> <f family="Arial Cyr" 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<ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">rows</ml:id> <ml:id xml:space="preserve">R1</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="82"/> </region> <region region-id="229" left="72" top="3588.75" width="81" height="15.75" align-x="125.25" align-y="3600" 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">rows</ml:id> <ml:id xml:space="preserve">R1</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>31</ml:real> </result> </ml:eval> </math> <rendering item-idref="83"/> </region> <region region-id="230" left="174" top="3588.75" width="70.5" height="15.75" align-x="222.75" align-y="3600" 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">cols</ml:id> <ml:id xml:space="preserve">R1</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </result> </ml:eval> </math> <rendering item-idref="84"/> </region> <region region-id="234" left="24" top="3624" width="350.25" height="191.25" align-x="24" align-y="3624" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="85"/> <rendering item-idref="86"/> </region> <region region-id="235" left="48" top="3840.75" width="431.25" height="30" align-x="48" 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="false"> <f family="Arial Cyr" charset="204" size="12">Из рис.3а видно, что метод Эйлера менее точен. Однако, с уменьшением диск-</f> <f family="Arial Cyr" charset="204" size="12">рета решение y(x)<sp count="2"/>стремится к R1</f> <f family="Arial Cyr" charset="204" size="12"> i,1 </f> <f family="Arial Cyr" charset="204" size="12">.</f> </text> </region> <region region-id="245" left="60" top="3876.75" width="440.25" height="13.5" align-x="60" align-y="3888" 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 2 </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr underline="false">.<sp count="3"/>Найти решение дифференциального уранения методом Эйлера:</inlineAttr> </f> </text> </region> <region region-id="246" left="192" top="3897" width="69.75" height="36.75" align-x="233.25" align-y="3918" 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:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">y</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="87"/> </region> <region region-id="248" left="48" top="3942.75" width="305.25" height="21.75" align-x="48" align-y="3960" 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">В упр.1 надо f(x,y) заменить на </f> <region region-id="249" left="216" 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