AA1-5.ptx

<exercise checkit-seed="0005" 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>
                              0 = -5 \, t {\frac{dx}{dt}} + 2 \, x {\frac{dx}{dt}} - 5 \, x;\qquad
                              x(-1)=
                              1                          </m>
                      </li>
                      <li>Solution: <m>-5 \, t x + x^{2} = 6</m></li>
                  </ul>
              </li>
              <li>
                  <ul>
                      <li>IVP:
                          <m>
                              5 \, y = 4 \, {\frac{dy}{dt}} + {\frac{d^2y}{dt^2}};\qquad
                              y(0)=
                              5,
                                  y'(0)=
                                  5
                          </m>
                      </li>
                      <li>Solution: <m>y = 5 \, e^{t}</m></li>
                  </ul>
              </li>
              <li>
                  <ul>
                      <li>IVP:
                          <m>
                              8 \, t^{4} - t {x'} = -2 \, x;\qquad
                              x(-1)=
                              0                          </m>
                      </li>
                      <li>Solution: <m>x = 4 \, t^{4} - 4 \, t^{2}</m></li>
                  </ul>
              </li>
      </ol>
  </statement>
  <answer>
      <ol>
              <li>
                  <ul>
                      <li>ODE: <m>0 = -5 \, t {\frac{dx}{dt}} + 2 \, x {\frac{dx}{dt}} - 5 \, x</m></li>
                      <li>IV(s): <m>
                          x(-1)=1
                          
                          </m></li>
                      <li>Order: 1st</li>
                      <li>Independent variable: <m>t</m></li>
                      <li>Dependent variable: <m>x</m></li>
                      <li>The solution <m>-5 \, t x + x^{2} = 6</m> is implicit.</li>
                  </ul>
              </li>
              <li>
                  <ul>
                      <li>ODE: <m>5 \, y = 4 \, {\frac{dy}{dt}} + {\frac{d^2y}{dt^2}}</m></li>
                      <li>IV(s): <m>
                          y(0)=5
                          ,y'(0)=5
                          </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 = 5 \, e^{t}</m> is explicit.</li>
                  </ul>
              </li>
              <li>
                  <ul>
                      <li>ODE: <m>8 \, t^{4} - t {x'} = -2 \, x</m></li>
                      <li>IV(s): <m>
                          x(-1)=0
                          
                          </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 = 4 \, t^{4} - 4 \, t^{2}</m> is explicit.</li>
                  </ul>
              </li>
      </ol>
  </answer>
</exercise>