0000.ptx

<exercise masterit-seed="3628" masterit-slug="D4" masterit-name="Using Laplace transforms to solve IVPs">
  <statement>
    <p>
          Explain how to solve the following IVP.
      </p>
    <me> 2 \, {y} = -2 \, {y''} - 2 \, \mathrm{u}\left(t - 3\right) \hspace{2em}
          y(0)= 0 ,
          y'(0)= -2 </me>
    <p>Hint: <m> \frac{1}{s^{3} + s} = -\frac{s}{s^{2} + 1} + \frac{1}{s} </m>.</p>
  </statement>
  <answer>
    <me>
          \mathcal{L}\{y\}= -\frac{2}{s^{2} + 1} - \frac{e^{\left(-3 \, s\right)}}{{\left(s^{2} + 1\right)} s} </me>
    <me>
          \mathcal{L}\{y\}= \frac{s e^{\left(-3 \, s\right)}}{s^{2} + 1} - \frac{e^{\left(-3 \, s\right)}}{s} - \frac{2}{s^{2} + 1} </me>
    <me> {y} = \cos\left(t - 3\right) \mathrm{u}\left(t - 3\right) - 2 \, \sin\left(t\right) - \mathrm{u}\left(t - 3\right) </me>
  </answer>
</exercise>