AA1-4.ptx

<exercise checkit-seed="0004" 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>
                              4 \, y^{3} {y'} + x {y'} + y = 0;\qquad
                              y(1)=
                              -1                          </m>
                      </li>
                      <li>Solution: <m>y^{4} + x y = 0</m></li>
                  </ul>
              </li>
              <li>
                  <ul>
                      <li>IVP:
                          <m>
                              0 = -16 \, y + {y''};\qquad
                              y(0)=
                              4,
                                  y'(0)=
                                  16
                          </m>
                      </li>
                      <li>Solution: <m>y = 4 \, e^{\left(4 \, t\right)}</m></li>
                  </ul>
              </li>
              <li>
                  <ul>
                      <li>IVP:
                          <m>
                              t {\frac{dy}{dt}} - 5 \, y = -4 \, t^{4};\qquad
                              y(-1)=
                              6                          </m>
                      </li>
                      <li>Solution: <m>y = -2 \, t^{5} + 4 \, t^{4}</m></li>
                  </ul>
              </li>
      </ol>
  </statement>
  <answer>
      <ol>
              <li>
                  <ul>
                      <li>ODE: <m>4 \, y^{3} {y'} + x {y'} + y = 0</m></li>
                      <li>IV(s): <m>
                          y(1)=-1
                          
                          </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^{4} + x y = 0</m> is implicit.</li>
                  </ul>
              </li>
              <li>
                  <ul>
                      <li>ODE: <m>0 = -16 \, y + {y''}</m></li>
                      <li>IV(s): <m>
                          y(0)=4
                          ,y'(0)=16
                          </m></li>
                      <li>Order: 2nd</li>
                      <li>Independent variable: <m>t</m></li>
                      <li>Dependent variable: <m>y</m></li>
                      <li>The solution <m>y = 4 \, e^{\left(4 \, t\right)}</m> is explicit.</li>
                  </ul>
              </li>
              <li>
                  <ul>
                      <li>ODE: <m>t {\frac{dy}{dt}} - 5 \, y = -4 \, t^{4}</m></li>
                      <li>IV(s): <m>
                          y(-1)=6
                          
                          </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 = -2 \, t^{5} + 4 \, t^{4}</m> is explicit.</li>
                  </ul>
              </li>
      </ol>
  </answer>
</exercise>