0002.ptx

<exercise masterit-seed="9524" 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> -3 \, t - {y'} = -y - \frac{3}{3 \, t - y - 3} </m>
      </li>
      <li>
        <m> t^{2} = -t y {y'} - y^{2} </m>
      </li>
    </ol>
  </statement>
  <answer>
    <ol>
      <li>
        <m> \frac{u {u'}}{u^{2} - 3} = 1 </m>
      </li>
      <li>
        <m> \frac{u {u'}}{u^{2} + 1} = \frac{1}{t} </m>
      </li>
    </ol>
  </answer>
</exercise>