0001.ptx

<exercise masterit-seed="7074" masterit-slug="D4" masterit-name="Using Laplace transforms to solve IVPs">
  <statement>
    <p>
          Explain how to solve the following IVP.
      </p>
    <me> -3 \, {y''} = 6 \, {y} - 9 \, {y'} - 3 \, \delta\left(t - 2\right) \hspace{2em}
          y(0)= 0 ,
          y'(0)= -2 </me>
    <p>Hint: <m> \frac{1}{s^{2} - 3 \, s + 2} = -\frac{1}{s - 1} + \frac{1}{s - 2} </m>.</p>
  </statement>
  <answer>
    <me>
          \mathcal{L}\{y\}= \frac{e^{\left(-2 \, s\right)}}{s^{2} - 3 \, s + 2} - \frac{2}{s^{2} - 3 \, s + 2} </me>
    <me>
          \mathcal{L}\{y\}= -\frac{e^{\left(-2 \, s\right)}}{s - 1} + \frac{e^{\left(-2 \, s\right)}}{s - 2} + \frac{2}{s - 1} - \frac{2}{s - 2} </me>
    <me> {y} = e^{\left(2 \, t - 4\right)} \mathrm{u}\left(t - 2\right) - e^{\left(t - 2\right)} \mathrm{u}\left(t - 2\right) - 2 \, e^{\left(2 \, t\right)} + 2 \, e^{t} </me>
  </answer>
</exercise>