常微分方程式をシンボリックに解く

シンボリック > シンボリック評価 > 常微分方程式をシンボリックに解く

常微分方程式をシンボリックに解く

solve キーワードを使用して、Mathcad でシンボリックに常微分方程式 (ODE) を解くことができます。

<region id="ID0E5IALB" actualWidth="344.33166666666671" actualHeight="114.152" top="96.000000000000014" left="19.200000000000003" xmlns="http://schemas.mathsoft.com/worksheet50">
  <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>

変更因子を関数の名前 y、または引数 y(x) を使用する関数として設定できます。

<region id="ID0E1JALB" actualWidth="342.56333333333339" actualHeight="114.152" top="288.00000000000006" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="11">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <equal />
        <apply>
          <plus />
          <apply>
            <apply>
              <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>

連立常微分方程式

<region id="ID0ERKALB" actualWidth="503.60666666666668" actualHeight="135.63133355712893" top="518.40000000000009" left="0" xmlns="http://schemas.mathsoft.com/worksheet50">
  <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>

連立常微分方程式および代数方程式を解くこともできます。

境界条件

• コーシー条件:

<region id="ID0EJNALB" actualWidth="260.875" actualHeight="70.96933355712892" top="748.80000000000007" left="57.600000000000009" xmlns="http://schemas.mathsoft.com/worksheet50">
  <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>

• 高階常微分方程式のノイマン条件:

<region id="ID0E2OALB" actualWidth="360.30666666666662" actualHeight="89.569333557128914" top="960.00000000000011" left="76.800000000000011" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="14">
    <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>

• 不等式条件:

<region id="ID0ENQALB" actualWidth="388.59833333333336" actualHeight="122.24666677856446" top="1152.0000000000002" left="0" xmlns="http://schemas.mathsoft.com/worksheet50">
  <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>

プライム演算子を使用した常微分方程式

• 微分、偏微分、およびプライム演算子を使用して、シンボリック評価のための常微分方程式を記述できます。

• 明示的に引数を指定せずにプライム演算子を使用します。この場合、Mathcad は自動的に変数を生成します。

<region id="ID0EOTALB" actualWidth="320.01000000000005" actualHeight="114.152" top="1344.0000000000002" left="57.600000000000009" xmlns="http://schemas.mathsoft.com/worksheet50">
  <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>

• Mathcad が微分変数を検出できない場合、次のエラーが表示されます。

部分解

generated キーワードを使用して、常微分方程式の部分解を受け取ります。

<region id="ID0E5VALB" actualWidth="454.32166666666672" actualHeight="53.743333445231123" top="1516.8000000000002" left="76.800000000000011" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="18">
    <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>
          <sequence>
            <id xml:space="preserve">solve</id>
            <id xml:space="preserve">y</id>
          </sequence>
          <apply>
            <equal />
            <id xml:space="preserve">generated</id>
            <id xml:space="preserve">complex</id>
          </apply>
        </sequence>
      </command>
      <symResult>
        <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>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
    <formattingOverride BackgroundColor="#00FFFFFF" />
  </math>
</region>

<region id="ID0EMWALB" actualWidth="451.14000000000004" actualHeight="97.738667114257822" top="1632.0000000000002" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="19">
    <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>
            <sequence>
              <id xml:space="preserve">solve</id>
              <id xml:space="preserve">y</id>
            </sequence>
            <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>