0005.ptx

<exercise masterit-seed="2484" masterit-slug="D4" masterit-name="Using Laplace transforms to solve IVPs">
  <statement>
    <p>
          Explain how to solve the following IVP.
      </p>
    <me> -3 \, {y''} = 18 \, {y} - 15 \, {y'} - 6 \, \delta\left(t - 3\right) \hspace{2em}
          y(0)= 0 ,
          y'(0)= 4 </me>
    <p>Hint: <m> \frac{1}{s^{2} - 5 \, s + 6} = -\frac{1}{s - 2} + \frac{1}{s - 3} </m>.</p>
  </statement>
  <answer>
    <me>
          \mathcal{L}\{y\}= \frac{2 \, e^{\left(-3 \, s\right)}}{s^{2} - 5 \, s + 6} + \frac{4}{s^{2} - 5 \, s + 6} </me>
    <me>
          \mathcal{L}\{y\}= -\frac{2 \, e^{\left(-3 \, s\right)}}{s - 2} + \frac{2 \, e^{\left(-3 \, s\right)}}{s - 3} - \frac{4}{s - 2} + \frac{4}{s - 3} </me>
    <me> {y} = 2 \, e^{\left(3 \, t - 9\right)} \mathrm{u}\left(t - 3\right) - 2 \, e^{\left(2 \, t - 6\right)} \mathrm{u}\left(t - 3\right) + 4 \, e^{\left(3 \, t\right)} - 4 \, e^{\left(2 \, t\right)} </me>
  </answer>
</exercise>