AA1-8.ptx

<exercise checkit-seed="0008" 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 + 4 \, {\frac{dx}{dt}} + {\frac{d^2x}{dt^2}} = 0;\qquad
                              x(0)=
                              3,
                                  x'(0)=
                                  -3
                          </m>
                      </li>
                      <li>Solution: <m>x = 3 \, e^{\left(-t\right)}</m></li>
                  </ul>
              </li>
              <li>
                  <ul>
                      <li>IVP:
                          <m>
                              -5 \, t {y'} - 5 \, y = -3 \, y^{2} {y'};\qquad
                              y(-1)=
                              1                          </m>
                      </li>
                      <li>Solution: <m>y^{3} - 5 \, t y = 6</m></li>
                  </ul>
              </li>
              <li>
                  <ul>
                      <li>IVP:
                          <m>
                              -10 \, t^{2} - t {x'} + 4 \, x = 0;\qquad
                              x(-1)=
                              2                          </m>
                      </li>
                      <li>Solution: <m>x = -3 \, t^{4} + 5 \, t^{2}</m></li>
                  </ul>
              </li>
      </ol>
  </statement>
  <answer>
      <ol>
              <li>
                  <ul>
                      <li>ODE: <m>3 \, x + 4 \, {\frac{dx}{dt}} + {\frac{d^2x}{dt^2}} = 0</m></li>
                      <li>IV(s): <m>
                          x(0)=3
                          ,x'(0)=-3
                          </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 = 3 \, e^{\left(-t\right)}</m> is explicit.</li>
                  </ul>
              </li>
              <li>
                  <ul>
                      <li>ODE: <m>-5 \, t {y'} - 5 \, y = -3 \, 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} - 5 \, t y = 6</m> is implicit.</li>
                  </ul>
              </li>
              <li>
                  <ul>
                      <li>ODE: <m>-10 \, t^{2} - t {x'} + 4 \, x = 0</m></li>
                      <li>IV(s): <m>
                          x(-1)=2
                          
                          </m></li>
                      <li>Order: 1st</li>
                      <li>Independent variable: <m>t</m></li>
                      <li>Dependent variable: <m>x</m></li>
                      <li>The solution <m>x = -3 \, t^{4} + 5 \, t^{2}</m> is explicit.</li>
                  </ul>
              </li>
      </ol>
  </answer>
</exercise>