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show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </ml:parens> </math> <rendering item-idref="17"/> </region> <region region-id="249" left="42" top="478.5" width="335.25" height="33.75" align-x="67.5" align-y="498" 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">z</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">λ</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:real>999</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">c</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">λ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>999</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">c</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:id xml:space="preserve">h4</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">ho</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="253" left="48" top="552.75" width="297" height="13.5" align-x="48" align-y="564" 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 графическую иллюстрацию решения СДУ</f> </text> </region> <region region-id="41" left="48" top="714" width="450.75" height="199.5" align-x="48" align-y="714" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="19"/> <rendering item-idref="20"/> </region> <region region-id="200" left="48" top="936.75" width="463.5" height="13.5" align-x="48" align-y="948" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">Рис.1. Расчетные кинетичкские зависимости флотационного процесса очистки воды.</inlineAttr> </f> </text> </region> <region region-id="201" left="54" top="984.75" width="444" height="27" align-x="54" align-y="996" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">Решение СДУ, показанное на рис.1, имеет следующий вектор коэффициентов </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">k = (k1, </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">k2, </f> <f family="Arial Cyr" charset="204" size="12">k3, </f> <f family="Arial Cyr" charset="204" size="12">k4, </f> <f family="Arial Cyr" charset="204" size="12">k5,</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal"> k6), имеющих следующие численные значения:</inlineAttr> </f> </text> </region> <region region-id="68" left="60" top="1036.5" width="69" height="33.75" align-x="80.25" align-y="1056" 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">k1</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-3</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="67" left="156" top="1039.5" width="66" height="21" align-x="176.25" align-y="1056" 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">k2</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-7</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="22"/> </region> <region region-id="66" left="246" top="1039.5" width="66" height="21" align-x="266.25" align-y="1056" 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">k3</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-4</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="255" left="438" top="1062.75" width="15.75" height="15.75" align-x="444.75" align-y="1074" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </ml:parens> </math> <rendering item-idref="24"/> </region> <region region-id="76" left="54" top="1087.5" width="66" height="21" align-x="74.25" align-y="1104" 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">k4</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-6</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="77" left="150" top="1087.5" width="66" height="21" align-x="170.25" align-y="1104" 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">k5</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-6</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="78" left="264" top="1084.5" width="69" height="33.75" align-x="284.25" align-y="1104" 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">k6</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-6</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="202" left="54" top="1122.75" width="456.75" height="54" align-x="54" align-y="1134" 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 проведения анализа решения в интерактивном режиме изменения параметров </f> <f family="Arial Cyr" charset="204" size="12">системы будем менять два из них, которые наиболее существенно влияют на её динамику. Возьмем коэффициенты k1 и<sp count="2"/>k2, хотя с таким же успехом можно взять</f> <f family="Arial Cyr" charset="204" size="12">и любые другие, и запишем их как анимационные.</f> </text> </region> <region region-id="146" left="48" top="1192.5" width="162" height="39" align-x="68.25" align-y="1212" 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">k1</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-3</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:real>20</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">FRAME</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="187" left="246" top="1192.5" width="153" height="39" align-x="266.25" align-y="1212" 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">k2</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-7</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>7</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">FRAME</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="254" left="456" top="1200.75" width="15.75" height="15.75" align-x="462.75" align-y="1212" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>3</ml:real> </ml:parens> </math> <rendering item-idref="30"/> </region> <region region-id="203" left="54" top="1242.75" width="453" height="27" align-x="54" align-y="1254" 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">Найдём в общем виде для заданных выше коэффициентов </f> <f family="Arial Cyr" charset="204" size="12">k (2)</f> <f family="Arial Cyr" charset="204" size="12"> решение СДУ, записывая </f> <f family="Arial Cyr" charset="204" size="12">следующие её параметры</f> </text> </region> <region region-id="87" left="54" top="1300.5" width="197.25" height="33.75" align-x="72.75" align-y="1320" 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">aa</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>-1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k1</ml:id> <ml:id xml:space="preserve">k2</ml:id> </ml:apply> <ml:id xml:space="preserve">k3</ml:id> </ml:apply> <ml:id xml:space="preserve">k4</ml:id> </ml:apply> <ml:id xml:space="preserve">k5</ml:id> </ml:apply> <ml:id xml:space="preserve">k6</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="31"/> </region> <region region-id="257" left="402" top="1308.75" width="15.75" height="15.75" align-x="408.75" align-y="1320" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>4</ml:real> </ml:parens> </math> <rendering item-idref="32"/> </region> <region region-id="258" left="498" top="1470.75" width="15.75" height="15.75" align-x="504.75" align-y="1482" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>5</ml:real> </ml:parens> </math> <rendering item-idref="33"/> </region> <region region-id="207" left="36" top="1410" width="447" height="92.25" align-x="56.25" align-y="1488" 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">bb</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">k1</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k2</ml:id> <ml:id xml:space="preserve">k3</ml:id> </ml:apply> <ml:id xml:space="preserve">k4</ml:id> </ml:apply> <ml:id xml:space="preserve">k5</ml:id> </ml:apply> <ml:id xml:space="preserve">k6</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">k2</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k3</ml:id> <ml:id xml:space="preserve">k4</ml:id> </ml:apply> <ml:id xml:space="preserve">k5</ml:id> </ml:apply> <ml:id xml:space="preserve">k6</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">k3</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k4</ml:id> <ml:id xml:space="preserve">k5</ml:id> </ml:apply> <ml:id xml:space="preserve">k6</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">k4</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">k5</ml:id> </ml:apply> <ml:id xml:space="preserve">k6</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">k5</ml:id> <ml:id xml:space="preserve">k6</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">k1</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">k2</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">k3</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">k4</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">k5</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">k6</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="259" left="42" top="1524.75" width="361.5" height="13.5" align-x="42" align-y="1536" 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">Тогда корни характеристического уравнения СДУ будут иметь вид</f> </text> </region> <region region-id="260" left="102" top="1560" width="73.5" height="18" align-x="129.75" align-y="1572" 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">λλ</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">aa</ml:id> <ml:id xml:space="preserve">bb</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="35"/> </region> <region region-id="261" left="246" top="1560" width="74.25" height="18" align-x="273.75" align-y="1572" 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">λλ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">aa</ml:id> <ml:id xml:space="preserve">bb</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="36"/> </region> <region region-id="262" left="432" top="1560.75" width="15.75" height="15.75" align-x="438.75" align-y="1572" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>6</ml:real> </ml:parens> </math> <rendering item-idref="37"/> </region> <region region-id="208" left="54" top="1590.75" width="438" height="27.75" align-x="54" align-y="1602" 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">Проведем проверку, сравнивая параметры СДУ рис.1. и найденные: аа = -а</f> <f family="Arial Cyr" charset="204" size="12">bb = b, </f> <f family="Symbol" charset="2" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">l</inlineAttr> </f> <f family="Symbol" charset="2" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">l</inlineAttr> </f> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">1 = </inlineAttr> </f> <f family="Symbol" charset="2" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">l</inlineAttr> </f> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">1, </inlineAttr> </f> <f family="Symbol" charset="2" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">l</inlineAttr> </f> <f family="Symbol" charset="2" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">l</inlineAttr> </f> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">2 = </inlineAttr> </f> <f family="Symbol" charset="2" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">l</inlineAttr> </f> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">2</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">. Таким образом</f> </text> </region> <region region-id="270" left="48" top="1633.5" width="105" height="21" align-x="66" align-y="1650" 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">aa</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.0003013</ml:real> </result> </ml:eval> </math> <rendering item-idref="38"/> </region> <region region-id="274" left="174" top="1633.5" width="93" height="21" align-x="186.75" align-y="1650" 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">a</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.0003013</ml:real> </result> </ml:eval> </math> <rendering item-idref="39"/> </region> <region region-id="276" left="294" top="1633.5" width="99.75" height="21" align-x="313.5" align-y="1650" 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">bb</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.00019970087631254898</ml:real> </result> </ml:eval> </math> <rendering item-idref="40"/> </region> <region region-id="278" left="414" top="1633.5" width="93.75" height="21" align-x="427.5" align-y="1650" 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">b</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.000199700876312549</ml:real> </result> </ml:eval> </math> <rendering item-idref="41"/> </region> <region region-id="271" left="48" top="1675.5" width="108" height="22.5" align-x="75" align-y="1692" 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">λλ</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.00050100087631254893</ml:real> </result> </ml:eval> </math> <rendering item-idref="42"/> </region> <region region-id="281" left="174" top="1675.5" width="114" height="22.5" align-x="201" align-y="1692" 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">λλ</ml:id> <ml:real>2</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.00010159912368745103</ml:real> </result> </ml:eval> </math> <rendering item-idref="43"/> </region> <region region-id="272" left="48" top="1711.5" width="107.25" height="22.5" align-x="68.25" align-y="1728" 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">λ</ml:id> <ml:real>2</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.000101599123687451</ml:real> </result> </ml:eval> </math> <rendering item-idref="44"/> </region> <region region-id="280" left="180" top="1711.5" width="101.25" height="22.5" align-x="200.25" align-y="1728" 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">λ</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.000501000876312549</ml:real> </result> </ml:eval> </math> <rendering item-idref="45"/> </region> <region region-id="230" left="60" top="1752.75" width="392.25" height="27" align-x="60" align-y="1764" 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">Получили тождества, что и требовалось показать. </f> <f family="Arial Cyr" charset="204" size="12">Запишем в общем (символьном) виде решение СДУ:</f> </text> </region> <region region-id="234" left="66" top="1810.5" width="166.5" height="25.5" align-x="97.5" 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:function> <ml:id xml:space="preserve">xx</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">λλ</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">λλ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="46"/> </region> <region region-id="233" left="66" top="1858.5" width="331.5" height="33.75" align-x="97.5" align-y="1878" 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">yy</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">λλ</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>-1001</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">c</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">λλ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:real>-1001</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">c</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:id xml:space="preserve">h4</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="47"/> </region> <region region-id="232" left="66" top="1930.5" width="351.75" height="33.75" align-x="96" align-y="1950" 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">zz</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">λλ</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:real>999</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">c</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">λλ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>999</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">c</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:id xml:space="preserve">h4</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">ho</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="48"/> </region> <region region-id="282" left="468" top="1938.75" width="15.75" height="15.75" align-x="474.75" align-y="1950" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>7</ml:real> </ml:parens> </math> <rendering item-idref="49"/> </region> <region region-id="235" left="66" top="1986.75" width="443.25" height="67.5" align-x="66" align-y="1998" 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">Решения xx(t), yy(t), zz(t) СДУ представим на рис.2., с которого снимем видеоклип,</f> <f family="Arial Cyr" charset="204" size="12">позволяющий увидеть динамику изменения качества системы в зависимости от</f> <f family="Arial Cyr" charset="204" size="12">изменения её параметров.</f> <f family="Arial Cyr" charset="204" size="12">Вычислим точку tm максимума на кривой yy(t) и некоторые другие параметры,</f> <f family="Arial Cyr" charset="204" size="12">которые также нанесём на рис.2.</f> </text> </region> <region region-id="240" left="60" top="2126.25" width="192.75" height="38.25" align-x="79.5" align-y="2148" 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">tm</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:real>-2</ml:real> <ml:id xml:space="preserve">bb</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ln</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">λλ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">λλ</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">c</ml:id> </ml:apply> <ml:real>1001</ml:real> </ml:apply> <ml:apply> <ml:minus/> <ml:real>1001</ml:real> <ml:apply> <ml:mult style="default"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">c</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="50"/> </region> <region region-id="241" left="294" top="2131.5" width="90.75" height="21" align-x="312.75" align-y="2148" 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">tm</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>4024.6810653936068</ml:real> </result> </ml:eval> </math> <rendering item-idref="51"/> </region> <region region-id="242" left="414" top="2110.5" width="64.5" height="51.75" align-x="428.25" align-y="2148" 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">tt</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">ln</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">k2</ml:id> <ml:id xml:space="preserve">k1</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k2</ml:id> <ml:id xml:space="preserve">k1</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="52"/> </region> <region region-id="243" left="30" top="2190" width="459" height="205.5" align-x="30" align-y="2190" 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="183" left="90" top="2413.5" width="85.5" height="21" align-x="103.5" align-y="2430" 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">tt</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>17037.7939416208</ml:real> </result> </ml:eval> </math> <rendering item-idref="55"/> </region> <region region-id="195" left="210" top="2413.5" width="90.75" height="21" align-x="228.75" align-y="2430" 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">tm</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>4024.6810653936068</ml:real> </result> </ml:eval> </math> <rendering item-idref="56"/> </region> <region region-id="244" left="54" top="2448.75" width="431.25" height="27" align-x="54" align-y="2460" 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">Рис.2. Расчетные кинетичкские зависимости флотационного процесса очистки </f> <f family="Arial Cyr" charset="204" size="12">воды и их анализ в режиме анимации.</f> </text> </region> <region region-id="284" left="60" top="2490.75" width="235.5" height="14.25" align-x="60" align-y="2502" 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"> Описание динамики качества системы. </f> </text> </region> <region region-id="287" left="48" top="2514.75" width="461.25" height="135" align-x="48" align-y="2526" 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">Концентрация флотокомплексов для СДУ с коэффициентами k, решения которой</f> <f family="Arial Cyr" charset="204" size="12">представлены на рис.1 и 2, достигает максимума при t = tm (8). Формула (8) </f> <f family="Arial Cyr" charset="204" size="12">выведена в общем виде и имеетсложную зависимость от коэффициентов (2) и</f> <f family="Arial Cyr" charset="204" size="12">некоторых других параметров (условий совместности уравнений СДУ и начальных</f> <f family="Arial Cyr" charset="204" size="12">(краевых) условий). Увидеть конкретные значения tm можно в режиме анимации,</f> <f family="Arial Cyr" charset="204" size="12">чтопозволяет оценить интервалы времени, когда флотокомплексы не успевают</f> <f family="Arial Cyr" charset="204" size="12">накапливаться и их концентрация в любой момент времени мала. Таким образом</f> <f family="Arial Cyr" charset="204" size="12">процесс флотации можно разделить на стадии. 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</math> <rendering item-idref="6"/> </region> <region region-id="246" left="390" top="126.75" width="67.5" height="15.75" align-x="402.75" align-y="138" 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">c</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>332.83479385424835</ml:real> </result> </ml:eval> </math> <rendering item-idref="7"/> </region> <region region-id="247" left="36" top="162.75" width="416.25" height="13.5" align-x="36" align-y="174" 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">2) корни характеристического уравнения и константы совмесности решений:</f> </text> </region> <region region-id="160" left="36" top="202.5" width="51" height="39" align-x="49.5" align-y="222" 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">a</ml:id> <ml:apply> <ml:div/> <ml:real>3013</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>7</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="161" left="114" top="202.5" width="54.75" height="39" align-x="128.25" align-y="222" 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:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">c</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>5</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>6</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="162" left="192" top="202.5" width="77.25" height="33.75" align-x="211.5" align-y="222" 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">h4</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>15020</ml:real> <ml:real>1003</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="163" left="294" top="202.5" width="89.25" height="33.75" align-x="313.5" align-y="222" 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">ho</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1018025</ml:real> <ml:real>1003</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="11"/> </region> <region region-id="164" left="402" top="205.5" width="93.75" height="21" align-x="415.5" align-y="222" 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">b</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.000199700876312549</ml:real> 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<ml:real>110779</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:matrix> </ml:symResult> </ml:symEval> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="18" left="330" top="282" width="63" height="18" align-x="351" 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:apply> <ml:indexer/> <ml:id xml:space="preserve">λ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">a</ml:id> </ml:apply> <ml:id xml:space="preserve">b</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="248" left="42" top="324.75" width="138.75" height="13.5" align-x="42" align-y="336" 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">3) запишем решения СДУ</f> </text> </region> <region region-id="251" left="42" top="358.5" width="149.25" height="25.5" align-x="68.25" 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:function> <ml:id xml:space="preserve">x</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id 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</ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>999</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">c</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:id xml:space="preserve">h4</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">ho</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="253" left="48" top="552.75" width="297" height="13.5" align-x="48" align-y="564" 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 графическую иллюстрацию решения СДУ</f> </text> </region> <region region-id="41" left="48" top="714" width="450.75" height="199.5" align-x="48" align-y="714" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="19"/> <rendering item-idref="20"/> </region> <region region-id="200" left="48" top="936.75" width="463.5" height="13.5" align-x="48" align-y="948" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">Рис.1. Расчетные кинетичкские зависимости флотационного процесса очистки воды.</inlineAttr> </f> </text> </region> <region region-id="201" left="54" top="984.75" width="444" height="27" align-x="54" align-y="996" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">Решение СДУ, показанное на рис.1, имеет следующий вектор коэффициентов </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">k = (k1, </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">k2, </f> <f family="Arial Cyr" charset="204" size="12">k3, </f> <f family="Arial Cyr" charset="204" size="12">k4, </f> <f family="Arial Cyr" charset="204" size="12">k5,</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal"> k6), имеющих следующие численные значения:</inlineAttr> </f> </text> </region> <region region-id="68" left="60" top="1036.5" width="69" height="33.75" align-x="80.25" align-y="1056" 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">k1</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-3</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="67" left="156" top="1039.5" width="66" height="21" align-x="176.25" align-y="1056" 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">k2</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-7</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="22"/> </region> <region region-id="66" left="246" top="1039.5" width="66" height="21" align-x="266.25" align-y="1056" 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">k3</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-4</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="255" left="438" top="1062.75" width="15.75" height="15.75" align-x="444.75" align-y="1074" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </ml:parens> </math> <rendering item-idref="24"/> </region> <region region-id="76" left="54" top="1087.5" width="66" height="21" align-x="74.25" align-y="1104" 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">k4</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-6</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="77" left="150" top="1087.5" width="66" height="21" align-x="170.25" align-y="1104" 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">k5</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-6</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="78" left="264" top="1084.5" width="69" height="33.75" align-x="284.25" align-y="1104" 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">k6</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-6</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="202" left="54" top="1122.75" width="456.75" height="54" align-x="54" align-y="1134" 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 проведения анализа решения в интерактивном режиме изменения параметров </f> <f family="Arial Cyr" charset="204" size="12">системы будем менять два из них, которые наиболее существенно влияют на её динамику. Возьмем коэффициенты k1 и<sp count="2"/>k2, хотя с таким же успехом можно взять</f> <f family="Arial Cyr" charset="204" size="12">и любые другие, и запишем их как анимационные.</f> </text> </region> <region region-id="146" left="48" top="1192.5" width="162" height="39" align-x="68.25" align-y="1212" 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">k1</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-3</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:real>20</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">FRAME</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="187" left="246" top="1192.5" width="153" height="39" align-x="266.25" align-y="1212" 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">k2</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-7</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>7</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">FRAME</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="254" left="456" top="1200.75" width="15.75" height="15.75" align-x="462.75" align-y="1212" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>3</ml:real> </ml:parens> </math> <rendering item-idref="30"/> </region> <region region-id="203" left="54" top="1242.75" width="453" height="27" align-x="54" align-y="1254" 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">Найдём в общем виде для заданных выше коэффициентов </f> <f family="Arial Cyr" charset="204" size="12">k (2)</f> <f family="Arial Cyr" charset="204" size="12"> решение СДУ, записывая </f> <f family="Arial Cyr" charset="204" size="12">следующие её параметры</f> </text> </region> <region region-id="87" left="54" top="1300.5" width="197.25" height="33.75" align-x="72.75" align-y="1320" 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">aa</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>-1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k1</ml:id> <ml:id xml:space="preserve">k2</ml:id> </ml:apply> <ml:id xml:space="preserve">k3</ml:id> </ml:apply> <ml:id xml:space="preserve">k4</ml:id> </ml:apply> <ml:id xml:space="preserve">k5</ml:id> </ml:apply> <ml:id xml:space="preserve">k6</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="31"/> </region> <region region-id="257" left="402" top="1308.75" width="15.75" height="15.75" align-x="408.75" align-y="1320" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>4</ml:real> </ml:parens> </math> <rendering item-idref="32"/> </region> <region region-id="258" left="498" top="1470.75" width="15.75" height="15.75" align-x="504.75" align-y="1482" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>5</ml:real> </ml:parens> </math> <rendering item-idref="33"/> </region> <region region-id="207" left="36" top="1410" width="447" height="92.25" align-x="56.25" align-y="1488" 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">bb</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">k1</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k2</ml:id> <ml:id xml:space="preserve">k3</ml:id> </ml:apply> <ml:id xml:space="preserve">k4</ml:id> </ml:apply> <ml:id xml:space="preserve">k5</ml:id> </ml:apply> <ml:id xml:space="preserve">k6</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">k2</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k3</ml:id> <ml:id xml:space="preserve">k4</ml:id> </ml:apply> <ml:id xml:space="preserve">k5</ml:id> </ml:apply> <ml:id xml:space="preserve">k6</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">k3</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k4</ml:id> <ml:id xml:space="preserve">k5</ml:id> </ml:apply> <ml:id xml:space="preserve">k6</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">k4</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">k5</ml:id> </ml:apply> <ml:id xml:space="preserve">k6</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">k5</ml:id> <ml:id xml:space="preserve">k6</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">k1</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">k2</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">k3</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">k4</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">k5</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">k6</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="259" left="42" top="1524.75" width="361.5" height="13.5" align-x="42" align-y="1536" 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">Тогда корни характеристического уравнения СДУ будут иметь вид</f> </text> </region> <region region-id="260" left="102" top="1560" width="73.5" height="18" align-x="129.75" align-y="1572" 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">λλ</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">aa</ml:id> <ml:id xml:space="preserve">bb</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="35"/> </region> <region region-id="261" left="246" top="1560" width="74.25" height="18" align-x="273.75" align-y="1572" 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">λλ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">aa</ml:id> <ml:id xml:space="preserve">bb</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="36"/> </region> <region region-id="262" left="432" top="1560.75" width="15.75" height="15.75" align-x="438.75" align-y="1572" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>6</ml:real> </ml:parens> </math> <rendering item-idref="37"/> </region> <region region-id="208" left="54" top="1590.75" width="438" height="27.75" align-x="54" align-y="1602" 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">Проведем проверку, сравнивая параметры СДУ рис.1. и найденные: аа = -а</f> <f family="Arial Cyr" charset="204" size="12">bb = b, </f> <f family="Symbol" charset="2" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">l</inlineAttr> </f> <f family="Symbol" charset="2" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">l</inlineAttr> </f> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">1 = </inlineAttr> </f> <f family="Symbol" charset="2" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">l</inlineAttr> </f> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">1, </inlineAttr> </f> <f family="Symbol" charset="2" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">l</inlineAttr> </f> <f family="Symbol" charset="2" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">l</inlineAttr> </f> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">2 = </inlineAttr> </f> <f family="Symbol" charset="2" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">l</inlineAttr> </f> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">2</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">. Таким образом</f> </text> </region> <region region-id="270" left="48" top="1633.5" width="105" height="21" align-x="66" align-y="1650" 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">aa</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.0003013</ml:real> </result> </ml:eval> </math> <rendering item-idref="38"/> </region> <region region-id="274" left="174" top="1633.5" width="93" height="21" align-x="186.75" align-y="1650" 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">a</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.0003013</ml:real> </result> </ml:eval> </math> <rendering item-idref="39"/> </region> <region region-id="276" left="294" top="1633.5" width="99.75" height="21" align-x="313.5" align-y="1650" 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">bb</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.00019970087631254898</ml:real> </result> </ml:eval> </math> <rendering item-idref="40"/> </region> <region region-id="278" left="414" top="1633.5" width="93.75" height="21" align-x="427.5" align-y="1650" 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">b</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.000199700876312549</ml:real> </result> </ml:eval> </math> <rendering item-idref="41"/> </region> <region region-id="271" left="48" top="1675.5" width="108" height="22.5" align-x="75" align-y="1692" 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">λλ</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.00050100087631254893</ml:real> </result> </ml:eval> </math> <rendering item-idref="42"/> </region> <region region-id="281" left="174" top="1675.5" width="114" height="22.5" align-x="201" align-y="1692" 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">λλ</ml:id> <ml:real>2</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.00010159912368745103</ml:real> </result> </ml:eval> </math> <rendering item-idref="43"/> </region> <region region-id="272" left="48" top="1711.5" width="107.25" height="22.5" align-x="68.25" align-y="1728" 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">λ</ml:id> <ml:real>2</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.000101599123687451</ml:real> </result> </ml:eval> </math> <rendering item-idref="44"/> </region> <region region-id="280" left="180" top="1711.5" width="101.25" height="22.5" align-x="200.25" align-y="1728" 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">λ</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.000501000876312549</ml:real> </result> </ml:eval> </math> <rendering item-idref="45"/> </region> <region region-id="230" left="60" top="1752.75" width="392.25" height="27" align-x="60" align-y="1764" 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">Получили тождества, что и требовалось показать. </f> <f family="Arial Cyr" charset="204" size="12">Запишем в общем (символьном) виде решение СДУ:</f> </text> </region> <region region-id="234" left="66" top="1810.5" width="166.5" height="25.5" align-x="97.5" 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:function> <ml:id xml:space="preserve">xx</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">λλ</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">λλ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="46"/> </region> <region region-id="233" left="66" top="1858.5" width="331.5" height="33.75" align-x="97.5" align-y="1878" 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">yy</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">λλ</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>-1001</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">c</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id 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height="15.75" align-x="474.75" align-y="1950" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>7</ml:real> </ml:parens> </math> <rendering item-idref="49"/> </region> <region region-id="235" left="66" top="1986.75" width="443.25" height="67.5" align-x="66" align-y="1998" 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">Решения xx(t), yy(t), zz(t) СДУ представим на рис.2., с которого снимем видеоклип,</f> <f family="Arial Cyr" charset="204" size="12">позволяющий увидеть динамику изменения качества системы в зависимости от</f> <f family="Arial Cyr" charset="204" size="12">изменения её параметров.</f> <f family="Arial Cyr" charset="204" size="12">Вычислим точку tm максимума на кривой yy(t) и некоторые другие параметры,</f> <f family="Arial Cyr" charset="204" size="12">которые также нанесём на рис.2.</f> </text> </region> <region region-id="240" left="60" top="2126.25" width="192.75" height="38.25" align-x="79.5" align-y="2148" 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">tm</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:real>-2</ml:real> <ml:id xml:space="preserve">bb</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ln</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> 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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">tm</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>4024.6810653936068</ml:real> </result> </ml:eval> </math> <rendering item-idref="51"/> </region> <region region-id="242" left="414" top="2110.5" width="64.5" height="51.75" align-x="428.25" align-y="2148" 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">tt</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">ln</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">k2</ml:id> <ml:id xml:space="preserve">k1</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k2</ml:id> <ml:id xml:space="preserve">k1</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="52"/> </region> <region region-id="243" left="30" top="2190" width="459" height="205.5" align-x="30" align-y="2190" 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="183" left="90" top="2413.5" width="85.5" height="21" align-x="103.5" align-y="2430" 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">tt</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>17037.7939416208</ml:real> </result> </ml:eval> </math> <rendering item-idref="55"/> </region> <region region-id="195" left="210" top="2413.5" width="90.75" height="21" align-x="228.75" align-y="2430" 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">tm</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>4024.6810653936068</ml:real> </result> </ml:eval> </math> <rendering item-idref="56"/> </region> <region region-id="244" left="54" top="2448.75" width="431.25" height="27" align-x="54" align-y="2460" 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">Рис.2. Расчетные кинетичкские зависимости флотационного процесса очистки </f> <f family="Arial Cyr" charset="204" size="12">воды и их анализ в режиме анимации.</f> </text> </region> <region region-id="284" left="60" top="2490.75" width="235.5" height="14.25" align-x="60" align-y="2502" 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"> Описание динамики качества системы. </f> </text> </region> <region region-id="287" left="48" top="2514.75" width="461.25" height="135" align-x="48" align-y="2526" 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">Концентрация флотокомплексов для СДУ с коэффициентами k, решения которой</f> <f family="Arial Cyr" charset="204" size="12">представлены на рис.1 и 2, достигает максимума при t = tm (8). Формула (8) </f> <f family="Arial Cyr" charset="204" size="12">выведена в общем виде и имеетсложную зависимость от коэффициентов (2) и</f> <f family="Arial Cyr" charset="204" size="12">некоторых других параметров (условий совместности уравнений СДУ и начальных</f> <f family="Arial Cyr" charset="204" size="12">(краевых) условий). Увидеть конкретные значения tm можно в режиме анимации,</f> <f family="Arial Cyr" charset="204" size="12">чтопозволяет оценить интервалы времени, когда флотокомплексы не успевают</f> <f family="Arial Cyr" charset="204" size="12">накапливаться и их концентрация в любой момент времени мала. Таким образом</f> <f family="Arial Cyr" charset="204" size="12">процесс флотации можно разделить на стадии. 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