0005.ptx

<exercise masterit-seed="4033" masterit-slug="F5" masterit-name="Substitution strategies">
  <statement>
    <p>
      Apply each of the substitutions <m>u=\frac{y}{t}</m> and
      <m>u=at+by+c</m> to one of the following
      ODEs, and show that each result is separable.
      (Do not fully solve either ODE.)
    </p>
    <ol>
      <li>
        <m> -t y {y'} + 2 \, t^{2} = y^{2} </m>
      </li>
      <li>
        <m> y - {y'} - 4 = 2 \, t - \frac{4}{2 \, t - y + 2} </m>
      </li>
    </ol>
  </statement>
  <answer>
    <ol>
      <li>
        <m> \frac{u {u'}}{u^{2} - 2} = \frac{1}{t} </m>
      </li>
      <li>
        <m> \frac{u {u'}}{u^{2} - 4} = 1 </m>
      </li>
    </ol>
  </answer>
</exercise>