Soluzione simbolica di equazioni differenziali ordinarie

Matematica simbolica > Valutazione simbolica > Soluzione simbolica di equazioni differenziali ordinarie

In Mathcad, è possibile risolvere simbolicamente le equazioni differenziali ordinarie (ODE) utilizzando la parola chiave solve.

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            <id xml:space="preserve">x</id>
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        <id xml:space="preserve">x</id>
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      <command>
        <sequence>
          <id xml:space="preserve">solve</id>
          <id xml:space="preserve">y</id>
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      </command>
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          <if>
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                <id xml:space="preserve" labels="CONSTANT">ℂ</id>
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            <then>
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                    <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>
                  <parens>
                    <apply>
                      <minus />
                      <id xml:space="preserve" labels="VARIABLE">x</id>
                      <real>1</real>
                    </apply>
                  </parens>
                </apply>
              </program>
            </then>
            <else>
              <program>
                <id xml:space="preserve" labels="*">undefined</id>
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    </symEval>
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Si può impostare il modificatore come nome y della funzione o come funzione con un argomento y(x).

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              <id xml:space="preserve">y</id>
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            <id xml:space="preserve">x</id>
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          <apply>
            <id xml:space="preserve">y</id>
            <id xml:space="preserve">x</id>
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        </apply>
        <id xml:space="preserve">x</id>
      </apply>
      <command>
        <sequence>
          <id xml:space="preserve">solve</id>
          <apply>
            <id xml:space="preserve">y</id>
            <id xml:space="preserve">x</id>
          </apply>
        </sequence>
      </command>
      <symResult>
        <program>
          <if>
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                <id xml:space="preserve" labels="VARIABLE">_z</id>
                <id xml:space="preserve" labels="CONSTANT">ℂ</id>
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            <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>
                  <parens>
                    <apply>
                      <minus />
                      <id xml:space="preserve" labels="VARIABLE">x</id>
                      <real>1</real>
                    </apply>
                  </parens>
                </apply>
              </program>
            </then>
            <else>
              <program>
                <id xml:space="preserve" labels="*">undefined</id>
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            </else>
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Un sistema di equazioni ODE

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                </lambda>
                <degree>
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          </apply>
          <real>1</real>
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          <equal />
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          <real>1</real>
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      <command>
        <sequence>
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            <id xml:space="preserve">solve</id>
            <id xml:space="preserve">y</id>
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          <id xml:space="preserve">z</id>
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                  <id xml:space="preserve" labels="VARIABLE">_z</id>
                  <id xml:space="preserve" labels="CONSTANT">ℂ</id>
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                <apply>
                  <elementOf />
                  <id xml:space="preserve" labels="VARIABLE">_z1</id>
                  <id xml:space="preserve" labels="CONSTANT">ℂ</id>
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            <then>
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                      <id xml:space="preserve" labels="CONSTANT">e</id>
                      <id xml:space="preserve" labels="VARIABLE">x</id>
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                  </apply>
                  <apply>
                    <plus />
                    <apply>
                      <neg />
                      <parens>
                        <apply>
                          <mult />
                          <id xml:space="preserve" labels="VARIABLE">_z1</id>
                          <apply>
                            <pow />
                            <id xml:space="preserve" labels="CONSTANT">e</id>
                            <id xml:space="preserve" labels="VARIABLE">x</id>
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                    <apply>
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È inoltre possibile risolvere un sistema di equazioni ODE ed equazioni algebriche.

Condizioni al limite

• Condizione di Cauchy:

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            <apply>
              <id xml:space="preserve">y</id>
              <id xml:space="preserve">x</id>
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          <real>1</real>
        </apply>
        <apply>
          <equal />
          <apply>
            <id xml:space="preserve">y</id>
            <real>0</real>
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          <real>2</real>
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      <command>
        <sequence>
          <id xml:space="preserve">solve</id>
          <id xml:space="preserve">y</id>
        </sequence>
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        <apply>
          <plus />
          <apply>
            <pow />
            <id xml:space="preserve" labels="CONSTANT">e</id>
            <apply>
              <neg />
              <id xml:space="preserve" labels="VARIABLE">x</id>
            </apply>
          </apply>
          <real>1</real>
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• Condizione di Neumann per ODE di ordine superiore:

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              </lambda>
              <degree>
                <real>2</real>
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            <apply>
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          </apply>
          <real>1</real>
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        <apply>
          <equal />
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            <id xml:space="preserve">y</id>
            <real>0</real>
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          <real>2</real>
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        <apply>
          <equal />
          <apply>
            <apply>
              <functionDerivative />
              <id xml:space="preserve">y</id>
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            <real>0</real>
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          <real>3</real>
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          <id xml:space="preserve">solve</id>
          <id xml:space="preserve">y</id>
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        <apply>
          <plus />
          <apply>
            <plus />
            <apply>
              <mult />
              <real>3</real>
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                <id xml:space="preserve" labels="VARIABLE">x</id>
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              <id xml:space="preserve" labels="VARIABLE">x</id>
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          <real>1</real>
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• Condizione di disuguaglianza:

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            <apply>
              <id xml:space="preserve">y</id>
              <id xml:space="preserve">x</id>
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          <real>1</real>
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        <apply>
          <greaterThan />
          <apply>
            <id xml:space="preserve">y</id>
            <id xml:space="preserve">x</id>
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          <real>0</real>
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          <id xml:space="preserve">solve</id>
          <id xml:space="preserve">y</id>
        </sequence>
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        <program>
          <if>
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                  <elementOf />
                  <id xml:space="preserve" labels="VARIABLE">x</id>
                  <id xml:space="preserve" labels="CONSTANT">ℝ</id>
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                <apply>
                  <greaterThan />
                  <apply>
                    <plus />
                    <apply>
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                      <id xml:space="preserve" labels="VARIABLE">_z</id>
                      <apply>
                        <pow />
                        <id xml:space="preserve" labels="CONSTANT">e</id>
                        <apply>
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                          <id xml:space="preserve" labels="VARIABLE">x</id>
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                    <real>1</real>
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                  <real>0</real>
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                  <real>1</real>
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ODE con operatore derivata prima

• È possibile utilizzare derivate, derivate parziali e operatori derivata prima per scrivere un'equazione ODE per la valutazione simbolica.

• Utilizzare l'operatore derivata prima senza specificare esplicitamente un argomento. Mathcad genera automaticamente la variabile nel caso riportato di seguito.

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                    <id xml:space="preserve" labels="CONSTANT">ℂ</id>
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                <apply>
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                    <id xml:space="preserve" labels="VARIABLE">y</id>
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                  <id xml:space="preserve" labels="CONSTANT">ℂ</id>
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            <then>
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                  <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>
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            <else>
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                <id xml:space="preserve" labels="*">undefined</id>
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• Se Mathcad non è in grado di rilevare la variabile differenziale, viene visualizzato un errore:

Argomenti correlati

Algoritmi per equazioni ODE

Risoluzione delle equazioni mediante i modificatori generati ed eval

Soluzione di ODE utilizzando il modificatore Generato

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