Аналитическое решение ОДУ

Символьные операции > Аналитические преобразования > Аналитическое решение ОДУ

Обыкновенные дифференциальные уравнения (ОДУ) можно решать в Mathcad аналитически с помощью ключевого слова solve.

Щелкните для копирования этого выражения

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                      <real>1</real>
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Модификатор можно задать как имя функции y или как функцию с аргументом y(x).

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        <id xml:space="preserve">x</id>
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      <command>
        <sequence>
          <id xml:space="preserve">solve</id>
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            <id xml:space="preserve">y</id>
            <id xml:space="preserve">x</id>
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                  <parens>
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                      <id xml:space="preserve" labels="VARIABLE">x</id>
                      <real>1</real>
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      </symResult>
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Система ОДУ

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                  <id xml:space="preserve" labels="CONSTANT">ℂ</id>
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                  <id xml:space="preserve" labels="VARIABLE">_z1</id>
                  <id xml:space="preserve" labels="CONSTANT">ℂ</id>
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                      <neg />
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Можно также решить систему ОДУ и алгебраическое уравнение.

Граничные условия

• Условие Коши:

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            <apply>
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          <real>1</real>
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        <apply>
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            <real>0</real>
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          <real>2</real>
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          <plus />
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            <id xml:space="preserve" labels="CONSTANT">e</id>
            <apply>
              <neg />
              <id xml:space="preserve" labels="VARIABLE">x</id>
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          <real>1</real>
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• Условие Неймана для ОДУ более высокого порядка:

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              </lambda>
              <degree>
                <real>2</real>
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            <apply>
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          <real>1</real>
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        <apply>
          <equal />
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            <real>0</real>
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          <real>2</real>
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        <apply>
          <equal />
          <apply>
            <apply>
              <functionDerivative />
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            <real>0</real>
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          <real>3</real>
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          <id xml:space="preserve">y</id>
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        <apply>
          <plus />
          <apply>
            <plus />
            <apply>
              <mult />
              <real>3</real>
              <apply>
                <id xml:space="preserve" labels="FUNCTION">sin</id>
                <id xml:space="preserve" labels="VARIABLE">x</id>
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            <apply>
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              <id xml:space="preserve" labels="VARIABLE">x</id>
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• Условие неравенства:

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              <id xml:space="preserve">x</id>
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          <id xml:space="preserve">y</id>
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                  <id xml:space="preserve" labels="VARIABLE">x</id>
                  <id xml:space="preserve" labels="CONSTANT">ℝ</id>
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ОДУ с оператором "Штрих"

• Можно использовать производные, частное производное и оператор "Штрих", чтобы записать ОДУ для аналитического преобразования.

• Используйте оператор "Штрих" без явного указания аргумента. В этом случае Mathcad генерирует переменную автоматически:

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            <id xml:space="preserve">y</id>
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        </apply>
        <real>1</real>
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      <command>
        <id xml:space="preserve">solve</id>
      </command>
      <symResult>
        <program>
          <if>
            <test>
              <apply>
                <and />
                <apply>
                  <and />
                  <apply>
                    <elementOf />
                    <id xml:space="preserve" labels="VARIABLE">_z</id>
                    <id xml:space="preserve" labels="CONSTANT">ℂ</id>
                  </apply>
                  <apply>
                    <elementOf />
                    <id xml:space="preserve" labels="VARIABLE">_x</id>
                    <id xml:space="preserve" labels="CONSTANT">ℂ</id>
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                </apply>
                <apply>
                  <elementOf />
                  <apply>
                    <id xml:space="preserve" labels="VARIABLE">y</id>
                    <id xml:space="preserve" labels="VARIABLE">_x</id>
                  </apply>
                  <id xml:space="preserve" labels="CONSTANT">ℂ</id>
                </apply>
              </apply>
            </test>
            <then>
              <program>
                <apply>
                  <plus />
                  <apply>
                    <mult />
                    <id xml:space="preserve" labels="VARIABLE">_z</id>
                    <apply>
                      <pow />
                      <id xml:space="preserve" labels="CONSTANT">e</id>
                      <apply>
                        <neg />
                        <id xml:space="preserve" labels="VARIABLE">_x</id>
                      </apply>
                    </apply>
                  </apply>
                  <real>1</real>
                </apply>
              </program>
            </then>
            <else>
              <program>
                <id xml:space="preserve" labels="*">undefined</id>
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        </program>
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• Если Mathcad не удается обнаружить дифференцируемую переменную, отображается сообщение об ошибке:

Частичные решения

Получите частичное решение ОДУ с помощью ключевого слова generated:

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            <id xml:space="preserve">complex</id>
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              <apply>
                <neg />
                <id xml:space="preserve" labels="VARIABLE">x</id>
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          <parens>
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              <id xml:space="preserve" labels="VARIABLE">x</id>
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            <id xml:space="preserve">z</id>
          </sequence>
          <apply>
            <equal />
            <id xml:space="preserve">generated</id>
            <apply>
              <id xml:space="preserve">eval</id>
              <real>0</real>
            </apply>
          </apply>
        </sequence>
      </command>
      <symResult>
        <matrix rows="1" cols="2">
          <real>0</real>
          <id xml:space="preserve" labels="VARIABLE">x</id>
        </matrix>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

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