AA1-6.ptx

<exercise checkit-seed="0006" checkit-slug="AA1" checkit-title="Structure of an IVP and Verifying Solutions">
  <statement>
      <p>
For each of the following Initial Value Problems (IVPs), designate the following:
      </p>
      <ul>
          <li>its Ordinary Differential Equation (ODE)</li>
          <li>its Initial Value or Values (IVs)</li>
          <li>the order of the IVP</li>
          <li>its independent variable</li>
          <li>its dependent variable</li>
          <li>whether its solution is implicit or explicit</li>
      </ul>
      <p>
Then show how to verify that its solution is valid.
      </p>
      <ol>
              <li>
                  <ul>
                      <li>IVP:
                          <m>
                              3 \, x^{3} = x {y'} - 2 \, y;\qquad
                              y(1)=
                              5                          </m>
                      </li>
                      <li>Solution: <m>y = 3 \, x^{3} + 2 \, x^{2}</m></li>
                  </ul>
              </li>
              <li>
                  <ul>
                      <li>IVP:
                          <m>
                              x - {\frac{d^2x}{dt^2}} = 0;\qquad
                              x(0)=
                              -2,
                                  x'(0)=
                                  2
                          </m>
                      </li>
                      <li>Solution: <m>x = -2 \, e^{\left(-t\right)}</m></li>
                  </ul>
              </li>
              <li>
                  <ul>
                      <li>IVP:
                          <m>
                              3 \, y^{2} {y'} - 2 \, t {y'} = 2 \, y;\qquad
                              y(-1)=
                              1                          </m>
                      </li>
                      <li>Solution: <m>y^{3} - 2 \, t y = 3</m></li>
                  </ul>
              </li>
      </ol>
  </statement>
  <answer>
      <ol>
              <li>
                  <ul>
                      <li>ODE: <m>3 \, x^{3} = x {y'} - 2 \, y</m></li>
                      <li>IV(s): <m>
                          y(1)=5
                          
                          </m></li>
                      <li>Order: 1st</li>
                      <li>Independent variable: <m>x</m></li>
                      <li>Dependent variable: <m>y</m></li>
                      <li>The solution <m>y = 3 \, x^{3} + 2 \, x^{2}</m> is explicit.</li>
                  </ul>
              </li>
              <li>
                  <ul>
                      <li>ODE: <m>x - {\frac{d^2x}{dt^2}} = 0</m></li>
                      <li>IV(s): <m>
                          x(0)=-2
                          ,x'(0)=2
                          </m></li>
                      <li>Order: 2nd</li>
                      <li>Independent variable: <m>t</m></li>
                      <li>Dependent variable: <m>x</m></li>
                      <li>The solution <m>x = -2 \, e^{\left(-t\right)}</m> is explicit.</li>
                  </ul>
              </li>
              <li>
                  <ul>
                      <li>ODE: <m>3 \, y^{2} {y'} - 2 \, t {y'} = 2 \, y</m></li>
                      <li>IV(s): <m>
                          y(-1)=1
                          
                          </m></li>
                      <li>Order: 1st</li>
                      <li>Independent variable: <m>t</m></li>
                      <li>Dependent variable: <m>y</m></li>
                      <li>The solution <m>y^{3} - 2 \, t y = 3</m> is implicit.</li>
                  </ul>
              </li>
      </ol>
  </answer>
</exercise>