0011.ptx

<exercise masterit-seed="7629" masterit-slug="D4" masterit-name="Using Laplace transforms to solve IVPs">
  <statement>
    <p>
          Explain how to solve the following IVP.
      </p>
    <me> -12 \, {y} - 3 \, {y''} - 12 \, \mathrm{u}\left(t - 2\right) = 0 \hspace{2em}
          y(0)= 5 ,
          y'(0)= 0 </me>
    <p>Hint: <m> \frac{1}{s^{3} + 4 \, s} = -\frac{s}{4 \, {\left(s^{2} + 4\right)}} + \frac{1}{4 \, s} </m>.</p>
  </statement>
  <answer>
    <me>
          \mathcal{L}\{y\}= \frac{5 \, s}{s^{2} + 4} - \frac{4 \, e^{\left(-2 \, s\right)}}{{\left(s^{2} + 4\right)} s} </me>
    <me>
          \mathcal{L}\{y\}= \frac{s e^{\left(-2 \, s\right)}}{s^{2} + 4} + \frac{5 \, s}{s^{2} + 4} - \frac{e^{\left(-2 \, s\right)}}{s} </me>
    <me> {y} = \cos\left(2 \, t - 4\right) \mathrm{u}\left(t - 2\right) + 5 \, \cos\left(2 \, t\right) - \mathrm{u}\left(t - 2\right) </me>
  </answer>
</exercise>