Solving ODEs Using the Generated Modifier

Symbolics > Symbolic Evaluation > Solving ODEs Using the Generated Modifier

Solving First Order ODE’s using Generated

Using the generated modifier for solving ODE’s allows you to set a range of particular solutions for the ODE, rather than receiving a general solution.

1. Solve the following ODE, that is written in prime notation, using the keyword 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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      </apply>
      <command>
        <sequence>
          <id xml:space="preserve">solve</id>
          <id xml:space="preserve">y</id>
        </sequence>
      </command>
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                <id xml:space="preserve" labels="VARIABLE">_z</id>
                <id xml:space="preserve" labels="CONSTANT">ℂ</id>
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            </test>
            <then>
              <program>
                <apply>
                  <minus />
                  <apply>
                    <mult />
                    <id xml:space="preserve" labels="VARIABLE">_z</id>
                    <apply>
                      <pow />
                      <id xml:space="preserve" labels="CONSTANT">e</id>
                      <id xml:space="preserve" labels="VARIABLE">x</id>
                    </apply>
                  </apply>
                  <parens>
                    <apply>
                      <plus />
                      <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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2. Solve the same ODE, this time adding the generated modifier, to receive a particular solution.

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            <id xml:space="preserve">y</id>
          </apply>
          <id xml:space="preserve">x</id>
        </apply>
        <apply>
          <plus />
          <apply>
            <id xml:space="preserve">y</id>
            <id xml:space="preserve">x</id>
          </apply>
          <id xml:space="preserve">x</id>
        </apply>
      </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>
            <apply>
              <id xml:space="preserve">eval</id>
              <real>0</real>
            </apply>
          </apply>
        </sequence>
      </command>
      <symResult>
        <apply>
          <minus />
          <apply>
            <neg />
            <id xml:space="preserve" labels="VARIABLE">x</id>
          </apply>
          <real>1</real>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
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3. You can solve ODE’s that are written in derivative notation using the generated modifier. You receive a particular solution.

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      <apply>
        <equal />
        <apply>
          <plus />
          <apply>
            <derivative />
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                <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>

Solving Second Order ODE’s using Generated

Use the generated modifier for solving second derivative ODE’s.

1. Solve the following second derivative ODE using the keyword solve.

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                  <id xml:space="preserve">s</id>
                </sequence>
              </apply>
            </lambda>
            <degree>
              <real>2</real>
            </degree>
          </apply>
          <apply>
            <mult />
            <id xml:space="preserve">a</id>
            <apply>
              <id xml:space="preserve">y</id>
              <sequence>
                <id xml:space="preserve">x</id>
                <id xml:space="preserve">s</id>
              </sequence>
            </apply>
          </apply>
        </apply>
        <id xml:space="preserve">b</id>
      </apply>
      <command>
        <sequence>
          <id xml:space="preserve">solve</id>
          <apply>
            <id xml:space="preserve">y</id>
            <sequence>
              <id xml:space="preserve">x</id>
              <id xml:space="preserve">s</id>
            </sequence>
          </apply>
        </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>
                <apply>
                  <plus />
                  <apply>
                    <mult />
                    <id xml:space="preserve" labels="VARIABLE">_z1</id>
                    <apply>
                      <pow />
                      <id xml:space="preserve" labels="CONSTANT">e</id>
                      <apply>
                        <div />
                        <apply>
                          <mult />
                          <id xml:space="preserve" labels="VARIABLE">x</id>
                          <apply>
                            <nthRoot />
                            <placeholder />
                            <apply>
                              <mult />
                              <real>4</real>
                              <id xml:space="preserve" labels="VARIABLE">a</id>
                            </apply>
                          </apply>
                        </apply>
                        <real>2</real>
                      </apply>
                    </apply>
                  </apply>
                  <parens>
                    <apply>
                      <minus />
                      <apply>
                        <mult />
                        <id xml:space="preserve" labels="VARIABLE">_z</id>
                        <apply>
                          <pow />
                          <id xml:space="preserve" labels="CONSTANT">e</id>
                          <apply>
                            <div />
                            <apply>
                              <neg />
                              <parens>
                                <apply>
                                  <mult />
                                  <id xml:space="preserve" labels="VARIABLE">x</id>
                                  <apply>
                                    <nthRoot />
                                    <placeholder />
                                    <apply>
                                      <mult />
                                      <real>4</real>
                                      <id xml:space="preserve" labels="VARIABLE">a</id>
                                    </apply>
                                  </apply>
                                </apply>
                              </parens>
                            </apply>
                            <real>2</real>
                          </apply>
                        </apply>
                      </apply>
                      <apply>
                        <div />
                        <id xml:space="preserve" labels="VARIABLE">b</id>
                        <id xml:space="preserve" labels="VARIABLE">a</id>
                      </apply>
                    </apply>
                  </parens>
                </apply>
              </program>
            </then>
            <else>
              <program>
                <id xml:space="preserve" labels="*">undefined</id>
              </program>
            </else>
          </if>
        </program>
      </symResult>
    </symEval>
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2. Solve the same second derivative ODE, this time adding the generated modifier. You receive a particular solution.

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                  <id xml:space="preserve">s</id>
                </sequence>
              </apply>
            </lambda>
            <degree>
              <real>2</real>
            </degree>
          </apply>
          <apply>
            <mult />
            <id xml:space="preserve">a</id>
            <apply>
              <id xml:space="preserve">y</id>
              <sequence>
                <id xml:space="preserve">x</id>
                <id xml:space="preserve">s</id>
              </sequence>
            </apply>
          </apply>
        </apply>
        <id xml:space="preserve">b</id>
      </apply>
      <command>
        <sequence>
          <sequence>
            <id xml:space="preserve">solve</id>
            <apply>
              <id xml:space="preserve">y</id>
              <sequence>
                <id xml:space="preserve">x</id>
                <id xml:space="preserve">s</id>
              </sequence>
            </apply>
          </sequence>
          <apply>
            <equal />
            <id xml:space="preserve">generated</id>
            <apply>
              <id xml:space="preserve">eval</id>
              <real>0</real>
            </apply>
          </apply>
        </sequence>
      </command>
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        <apply>
          <neg />
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            <id xml:space="preserve" labels="VARIABLE">b</id>
            <id xml:space="preserve" labels="VARIABLE">a</id>
          </apply>
        </apply>
      </symResult>
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Solving a System of ODE’s using Generated

Solve a system of ODE’s using the generated modifier, receiving a particular solution.

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              </lambda>
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                <placeholder />
              </degree>
            </apply>
            <parens>
              <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>
            </parens>
          </apply>
          <real>1</real>
        </apply>
        <apply>
          <equal />
          <apply>
            <plus />
            <apply>
              <derivative />
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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>
                </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>
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      <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>
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