AA1-9.ptx

<exercise checkit-seed="0009" 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>
                              2 \, y {y'} = -4 \, t {y'} - 4 \, y;\qquad
                              y(1)=
                              -1                          </m>
                      </li>
                      <li>Solution: <m>4 \, t y + y^{2} = \left(-3\right)</m></li>
                  </ul>
              </li>
              <li>
                  <ul>
                      <li>IVP:
                          <m>
                              -x {y'} = x^{3} - 2 \, y;\qquad
                              y(-1)=
                              3                          </m>
                      </li>
                      <li>Solution: <m>y = -x^{3} + 2 \, x^{2}</m></li>
                  </ul>
              </li>
              <li>
                  <ul>
                      <li>IVP:
                          <m>
                              0 = -5 \, y - 4 \, {\frac{dy}{dt}} + {\frac{d^2y}{dt^2}};\qquad
                              y(0)=
                              -3,
                                  y'(0)=
                                  -15
                          </m>
                      </li>
                      <li>Solution: <m>y = -3 \, e^{\left(5 \, t\right)}</m></li>
                  </ul>
              </li>
      </ol>
  </statement>
  <answer>
      <ol>
              <li>
                  <ul>
                      <li>ODE: <m>2 \, y {y'} = -4 \, t {y'} - 4 \, 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>4 \, t y + y^{2} = \left(-3\right)</m> is implicit.</li>
                  </ul>
              </li>
              <li>
                  <ul>
                      <li>ODE: <m>-x {y'} = x^{3} - 2 \, y</m></li>
                      <li>IV(s): <m>
                          y(-1)=3
                          
                          </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 = -x^{3} + 2 \, x^{2}</m> is explicit.</li>
                  </ul>
              </li>
              <li>
                  <ul>
                      <li>ODE: <m>0 = -5 \, y - 4 \, {\frac{dy}{dt}} + {\frac{d^2y}{dt^2}}</m></li>
                      <li>IV(s): <m>
                          y(0)=-3
                          ,y'(0)=-15
                          </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 = -3 \, e^{\left(5 \, t\right)}</m> is explicit.</li>
                  </ul>
              </li>
      </ol>
  </answer>
</exercise>