기호 연산을 사용한 ODE 풀이

기호 > 기호 연산 > 기호 연산을 사용한 ODE 풀이

기호 연산을 사용한 ODE 풀이

Mathcad에서 solve 키워드를 사용하여 ODE(상미분 방정식)를 기호 연산으로 풀 수 있습니다.

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  <math resultRef="10">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <equal />
        <apply>
          <plus />
          <apply>
            <derivative />
            <lambda>
              <boundVars>
                <id xml:space="preserve">x</id>
              </boundVars>
              <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>
        <id xml:space="preserve">x</id>
      </apply>
      <command>
        <sequence>
          <id xml:space="preserve">solve</id>
          <id xml:space="preserve">y</id>
        </sequence>
      </command>
      <symResult>
        <program>
          <if>
            <test>
              <apply>
                <elementOf />
                <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="*">undefined</id>
              </program>
            </else>
          </if>
        </program>
      </symResult>
    </symEval>
    <formattingOverride BackgroundColor="#00FFFFFF" />
  </math>
</region>

수정자는 함수의 이름 y 또는 인수가 있는 함수y(x)로 설정할 수 있습니다.

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    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <equal />
        <apply>
          <plus />
          <apply>
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              <functionDerivative />
              <id xml:space="preserve">y</id>
            </apply>
            <id xml:space="preserve">x</id>
          </apply>
          <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>
      <symResult>
        <program>
          <if>
            <test>
              <apply>
                <elementOf />
                <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="*">undefined</id>
              </program>
            </else>
          </if>
        </program>
      </symResult>
    </symEval>
  </math>
</region>

ODE 시스템

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  <math resultRef="12">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <matrix rows="2" cols="1">
        <apply>
          <equal />
          <apply>
            <plus />
            <apply>
              <derivative />
              <lambda>
                <boundVars>
                  <id xml:space="preserve">x</id>
                </boundVars>
                <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>
                </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>
            <apply>
              <id xml:space="preserve">y</id>
              <id xml:space="preserve">x</id>
            </apply>
          </apply>
          <real>1</real>
        </apply>
      </matrix>
      <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>
              <program>
                <matrix rows="1" cols="2">
                  <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>
                  <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>
                    </apply>
                  </apply>
                </matrix>
              </program>
            </then>
            <else>
              <program>
                <id xml:space="preserve" labels="*">undefined</id>
              </program>
            </else>
          </if>
        </program>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

ODE 및 대수 방정식의 시스템을 풀 수도 있습니다.

경계 조건

• 코쉬 조건:

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  <math resultRef="13">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <matrix rows="2" cols="1">
        <apply>
          <equal />
          <apply>
            <plus />
            <apply>
              <derivative />
              <lambda>
                <boundVars>
                  <id xml:space="preserve">x</id>
                </boundVars>
                <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>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

• 고차 ODE의 노이만 조건:

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    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <matrix rows="3" cols="1">
        <apply>
          <equal />
          <apply>
            <plus />
            <apply>
              <derivative />
              <lambda>
                <boundVars>
                  <id xml:space="preserve">x</id>
                </boundVars>
                <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>
        </apply>
      </matrix>
      <command>
        <sequence>
          <id xml:space="preserve">solve</id>
          <id xml:space="preserve">y</id>
        </sequence>
      </command>
      <symResult>
        <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>
              </apply>
            </apply>
            <apply>
              <id xml:space="preserve" labels="FUNCTION">cos</id>
              <id xml:space="preserve" labels="VARIABLE">x</id>
            </apply>
          </apply>
          <real>1</real>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

• 부등식 조건:

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  <math resultRef="15">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <matrix rows="2" cols="1">
        <apply>
          <equal />
          <apply>
            <plus />
            <apply>
              <derivative />
              <lambda>
                <boundVars>
                  <id xml:space="preserve">x</id>
                </boundVars>
                <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>
      </matrix>
      <command>
        <sequence>
          <id xml:space="preserve">solve</id>
          <id xml:space="preserve">y</id>
        </sequence>
      </command>
      <symResult>
        <program>
          <if>
            <test>
              <apply>
                <and />
                <apply>
                  <elementOf />
                  <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>
                      </apply>
                    </apply>
                    <real>1</real>
                  </apply>
                  <real>0</real>
                </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>
              </program>
            </else>
          </if>
        </program>
      </symResult>
    </symEval>
  </math>
</region>

프라임 연산자가 있는 ODE

• 도함수, 편도함수, 프라임 연산자를 사용하여 기호 연산을 할 ODE를 작성할 수 있습니다.

• 인수를 명시적으로 지정하지 않고 프라임 연산자를 사용합니다. 이 경우 Mathcad가 변수를 자동으로 생성합니다.

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  <math resultRef="16">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <equal />
        <apply>
          <plus />
          <id xml:space="preserve">y</id>
          <apply>
            <functionDerivative />
            <id xml:space="preserve">y</id>
          </apply>
        </apply>
        <real>1</real>
      </apply>
      <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>
                  </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="*">undefined</id>
              </program>
            </else>
          </if>
        </program>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

• Mathcad가 미분 변수를 감지할 수 없는 경우 다음과 같은 오류가 표시됩니다.