以符号形式求解 ODE

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以符号形式求解 ODE

在 Mathcad 中，可使用关键字 solve 以符号形式求解常微分方程 (ODE)。

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            <id xml:space="preserve">x</id>
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        <id xml:space="preserve">x</id>
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        <sequence>
          <id xml:space="preserve">solve</id>
          <id xml:space="preserve">y</id>
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      <symResult>
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          <if>
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                <elementOf />
                <id xml:space="preserve" labels="VARIABLE">_z</id>
                <id xml:space="preserve" labels="CONSTANT">ℂ</id>
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            <then>
              <program>
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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>
                  <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>
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可将修饰符设置为函数名称 y，或设置为带有自变量 y(x) 的函数。

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            <id xml:space="preserve">x</id>
          </apply>
        </apply>
        <id xml:space="preserve">x</id>
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      <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>
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                <id xml:space="preserve" labels="*">undefined</id>
              </program>
            </else>
          </if>
        </program>
      </symResult>
    </symEval>
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ODE 方程组

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            <parens>
              <apply>
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                    <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>
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                </lambda>
                <degree>
                  <placeholder />
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          <real>1</real>
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          <equal />
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              <id xml:space="preserve">y</id>
              <id xml:space="preserve">x</id>
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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>
        </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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                      <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>
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                      </parens>
                    </apply>
                    <apply>
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                      <id xml:space="preserve" labels="VARIABLE">x</id>
                      <id xml:space="preserve" labels="VARIABLE">_z</id>
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也可求解 ODE 方程组和代数方程组。

边界条件

• 柯西边界条件：

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                </apply>
              </lambda>
              <degree>
                <placeholder />
              </degree>
            </apply>
            <apply>
              <id xml:space="preserve">y</id>
              <id xml:space="preserve">x</id>
            </apply>
          </apply>
          <real>1</real>
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        <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>
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• 高阶 ODE 的诺伊曼边界条件：

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                  <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
                </apply>
              </lambda>
              <degree>
                <real>2</real>
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            </apply>
            <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>
          <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>
            </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>
        </sequence>
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      <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>
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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>
        </apply>
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• 不等式边界条件：

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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 />
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            <id xml:space="preserve">x</id>
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          <real>0</real>
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      <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>
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                  <real>1</real>
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                <id xml:space="preserve" labels="*">undefined</id>
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带有导数运算符的 ODE

• 要以符号形式求解 ODE，可用导数、偏导数和导数运算符来构成 ODE。

• 使用导数运算符但不明确指定自变量。此时，Mathcad 会自动生成变量：

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        <real>1</real>
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      <command>
        <id xml:space="preserve">solve</id>
      </command>
      <symResult>
        <program>
          <if>
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                <and />
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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">_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>
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              </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>
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            </then>
            <else>
              <program>
                <id xml:space="preserve" labels="*">undefined</id>
              </program>
            </else>
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        </program>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
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• 如果 Mathcad 无法检测到微分变量，则会显示错误：