0004.ptx

<exercise masterit-seed="5049" masterit-slug="D4" masterit-name="Using Laplace transforms to solve IVPs">
  <statement>
    <p>
          Explain how to solve the following IVP.
      </p>
    <me> -12 \, {y} + 12 \, \delta\left(t - 1\right) = -3 \, {y''} \hspace{2em}
          y(0)= 0 ,
          y'(0)= -8 </me>
    <p>Hint: <m> \frac{1}{s^{2} - 4} = -\frac{1}{4 \, {\left(s + 2\right)}} + \frac{1}{4 \, {\left(s - 2\right)}} </m>.</p>
  </statement>
  <answer>
    <me>
          \mathcal{L}\{y\}= -\frac{4 \, e^{\left(-s\right)}}{s^{2} - 4} - \frac{8}{s^{2} - 4} </me>
    <me>
          \mathcal{L}\{y\}= \frac{e^{\left(-s\right)}}{s + 2} - \frac{e^{\left(-s\right)}}{s - 2} + \frac{2}{s + 2} - \frac{2}{s - 2} </me>
    <me> {y} = -e^{\left(2 \, t - 2\right)} \mathrm{u}\left(t - 1\right) + e^{\left(-2 \, t + 2\right)} \mathrm{u}\left(t - 1\right) - 2 \, e^{\left(2 \, t\right)} + 2 \, e^{\left(-2 \, t\right)} </me>
  </answer>
</exercise>