GDG symbolisch lösen

Symbolische Mathematik > Symbolische Auswertung > GDG symbolisch lösen

Sie können gewöhnliche Differentialgleichungen symbolisch auf Mathcad lösen, indem Sie das Schlüsselwort solve verwenden.

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            <id xml:space="preserve">x</id>
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        <id xml:space="preserve">x</id>
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          <id xml:space="preserve">solve</id>
          <id xml:space="preserve">y</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>
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Sie können den Modifizierer als Funktionsnamen y oder als Funktion mit einem Argument y(x) angeben.

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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>
          </apply>
        </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>
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        <program>
          <if>
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                <id xml:space="preserve" labels="VARIABLE">_z</id>
                <id xml:space="preserve" labels="CONSTANT">ℂ</id>
              </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>
                  <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="*">nicht definiert</id>
              </program>
            </else>
          </if>
        </program>
      </symResult>
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Ein System von GDG

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                <apply>
                  <id xml:space="preserve">y</id>
                  <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
                </apply>
              </lambda>
              <degree>
                <placeholder />
              </degree>
            </apply>
            <parens>
              <apply>
                <derivative />
                <lambda>
                  <boundVars>
                    <id xml:space="preserve">x</id>
                  </boundVars>
                  <apply>
                    <id xml:space="preserve">z</id>
                    <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
                  </apply>
                </lambda>
                <degree>
                  <placeholder />
                </degree>
              </apply>
            </parens>
          </apply>
          <real>1</real>
        </apply>
        <apply>
          <equal />
          <apply>
            <plus />
            <apply>
              <derivative />
              <lambda>
                <boundVars>
                  <id xml:space="preserve">x</id>
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                <apply>
                  <id xml:space="preserve">z</id>
                  <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
                </apply>
              </lambda>
              <degree>
                <placeholder />
              </degree>
            </apply>
            <apply>
              <id xml:space="preserve">y</id>
              <id xml:space="preserve">x</id>
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          </apply>
          <real>1</real>
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      <command>
        <sequence>
          <sequence>
            <id xml:space="preserve">solve</id>
            <id xml:space="preserve">y</id>
          </sequence>
          <id xml:space="preserve">z</id>
        </sequence>
      </command>
      <symResult>
        <program>
          <if>
            <test>
              <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">_z1</id>
                  <id xml:space="preserve" labels="CONSTANT">ℂ</id>
                </apply>
              </apply>
            </test>
            <then>
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                    <mult />
                    <id xml:space="preserve" labels="VARIABLE">_z1</id>
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                      <pow />
                      <id xml:space="preserve" labels="CONSTANT">e</id>
                      <id xml:space="preserve" labels="VARIABLE">x</id>
                    </apply>
                  </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>
                          </apply>
                        </apply>
                      </parens>
                    </apply>
                    <apply>
                      <plus />
                      <id xml:space="preserve" labels="VARIABLE">x</id>
                      <id xml:space="preserve" labels="VARIABLE">_z</id>
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Sie können auch ein System aus GDG und algebraischen Gleichungen lösen.

Randbedingungen

• Cauchy-Bedingung:

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                <apply>
                  <id xml:space="preserve">y</id>
                  <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
                </apply>
              </lambda>
              <degree>
                <placeholder />
              </degree>
            </apply>
            <apply>
              <id xml:space="preserve">y</id>
              <id xml:space="preserve">x</id>
            </apply>
          </apply>
          <real>1</real>
        </apply>
        <apply>
          <equal />
          <apply>
            <id xml:space="preserve">y</id>
            <real>0</real>
          </apply>
          <real>2</real>
        </apply>
      </matrix>
      <command>
        <sequence>
          <id xml:space="preserve">solve</id>
          <id xml:space="preserve">y</id>
        </sequence>
      </command>
      <symResult>
        <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>
        </apply>
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• Neumann-Bedingung für GDG höherer Ordnung:

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                <apply>
                  <id xml:space="preserve">y</id>
                  <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
                </apply>
              </lambda>
              <degree>
                <real>2</real>
              </degree>
            </apply>
            <apply>
              <id xml:space="preserve">y</id>
              <id xml:space="preserve">x</id>
            </apply>
          </apply>
          <real>1</real>
        </apply>
        <apply>
          <equal />
          <apply>
            <id xml:space="preserve">y</id>
            <real>0</real>
          </apply>
          <real>2</real>
        </apply>
        <apply>
          <equal />
          <apply>
            <apply>
              <functionDerivative />
              <id xml:space="preserve">y</id>
            </apply>
            <real>0</real>
          </apply>
          <real>3</real>
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      <command>
        <sequence>
          <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>
              <apply>
                <id xml:space="preserve" labels="FUNCTION">sin</id>
                <id xml:space="preserve" labels="VARIABLE">x</id>
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            <apply>
              <id xml:space="preserve" labels="FUNCTION">cos</id>
              <id xml:space="preserve" labels="VARIABLE">x</id>
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          <real>1</real>
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• Ungleichheitsbedingung:

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                <apply>
                  <id xml:space="preserve">y</id>
                  <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
                </apply>
              </lambda>
              <degree>
                <placeholder />
              </degree>
            </apply>
            <apply>
              <id xml:space="preserve">y</id>
              <id xml:space="preserve">x</id>
            </apply>
          </apply>
          <real>1</real>
        </apply>
        <apply>
          <greaterThan />
          <apply>
            <id xml:space="preserve">y</id>
            <id xml:space="preserve">x</id>
          </apply>
          <real>0</real>
        </apply>
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      <command>
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          <id xml:space="preserve">solve</id>
          <id xml:space="preserve">y</id>
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        <program>
          <if>
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                  <id xml:space="preserve" labels="VARIABLE">x</id>
                  <id xml:space="preserve" labels="CONSTANT">ℝ</id>
                </apply>
                <apply>
                  <greaterThan />
                  <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>
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                    <real>1</real>
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                  <real>0</real>
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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>
                  <real>1</real>
                </apply>
              </program>
            </then>
            <else>
              <program>
                <id xml:space="preserve" labels="*">nicht definiert</id>
              </program>
            </else>
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        </program>
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GDG mit dem Operator für gewöhnliche Ableitungen

• Sie können Ableitungen, partielle Ableitungen und den Operator für gewöhnliche Ableitungen verwenden, um eine GDG für die symbolische Auswertung zu schreiben.

• Verwenden Sie den Operator für gewöhnliche Ableitungen, ohne explizit ein Argument anzugeben. Mathcad generiert die Variable in diesem Fall automatisch:

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            <id xml:space="preserve">y</id>
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        <real>1</real>
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      <command>
        <id xml:space="preserve">solve</id>
      </command>
      <symResult>
        <program>
          <if>
            <test>
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                <and />
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                  <and />
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                    <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>
                  </apply>
                </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="*">nicht definiert</id>
              </program>
            </else>
          </if>
        </program>
      </symResult>
    </symEval>
    <resultFormat>
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• Wenn Mathcad die Differentialvariable nicht erkennen kann, wird ein Fehler angezeigt:

Zugehörige Themen

Algorithmen für gewöhnliche Differentialgleichungen

Gleichungen mithilfe der Modifizierer generated und eval lösen

GDGs mit dem Modifizierer "generated" lösen

War dies hilfreich?