0009.ptx

<exercise masterit-seed="2645" masterit-slug="D4" masterit-name="Using Laplace transforms to solve IVPs">
  <statement>
    <p>
          Explain how to solve the following IVP.
      </p>
    <me> 3 \, {y''} = -27 \, {y} - 81 \, \mathrm{u}\left(t - 2\right) \hspace{2em}
          y(0)= -2 ,
          y'(0)= 0 </me>
    <p>Hint: <m> \frac{1}{s^{3} + 9 \, s} = -\frac{s}{9 \, {\left(s^{2} + 9\right)}} + \frac{1}{9 \, s} </m>.</p>
  </statement>
  <answer>
    <me>
          \mathcal{L}\{y\}= -\frac{2 \, s}{s^{2} + 9} - \frac{27 \, e^{\left(-2 \, s\right)}}{{\left(s^{2} + 9\right)} s} </me>
    <me>
          \mathcal{L}\{y\}= \frac{3 \, s e^{\left(-2 \, s\right)}}{s^{2} + 9} - \frac{2 \, s}{s^{2} + 9} - \frac{3 \, e^{\left(-2 \, s\right)}}{s} </me>
    <me> {y} = 3 \, \cos\left(3 \, t - 6\right) \mathrm{u}\left(t - 2\right) - 2 \, \cos\left(3 \, t\right) - 3 \, \mathrm{u}\left(t - 2\right) </me>
  </answer>
</exercise>