AA5-3.ptx

<exercise checkit-seed="0003" checkit-slug="AA5" checkit-title="Strategies for Solving IVPs">
    <statement>
        <p>
For each ODE, describe an appropriate strategy to find its
general solution, and the features of the ODE that make
that strategy appropriate. (Do not fully solve these ODEs.)
        </p>
        <ol>
                <li>
                    <m>-81 \, {y} = -{y''} + 3 \, \mathrm{u}\left(t - 5\right)</m>
                </li>
                <li>
                    <m>-8 \, {y}^{3} {y'} - 3 \, {y}^{2} = 6 \, {y} {y'} t + 4 \, t^{3}</m>
                </li>
                <li>
                    <m>{y'} t - 5 \, t^{2} = -3 \, {y}</m>
                </li>
                <li>
                    <m>-5 \, {y} + {y''} - 4 \, {y'} = 0</m>
                </li>
        </ol>
  </statement>
  <answer>
        <ol>
                <li>
The ODE is linear constant-coefficient with a discontinuous function, so it can be solved by using Laplace transforms.
                </li>
                <li>
The ODE is exact, so it can be solved by finding a potential function.
                </li>
                <li>
The ODE is linear first-order, so it can be solved by solving its homogeneous form and then using variation of parameters, or using an integrating factor.
                </li>
                <li>
The ODE is linear homogeneous with constant coefficients, so it can be solved by using D-notation and factoring.
                </li>
        </ol>
  </answer>
</exercise>