0002.ptx

<exercise masterit-seed="1525" masterit-slug="D4" masterit-name="Using Laplace transforms to solve IVPs">
  <statement>
    <p>
          Explain how to solve the following IVP.
      </p>
    <me> 4 \, {y} + 2 \, {y''} + 6 \, {y'} - 6 \, \delta\left(t - 3\right) = 0 \hspace{2em}
          y(0)= 0 ,
          y'(0)= 3 </me>
    <p>Hint: <m> \frac{1}{s^{2} + 3 \, s + 2} = -\frac{1}{s + 2} + \frac{1}{s + 1} </m>.</p>
  </statement>
  <answer>
    <me>
          \mathcal{L}\{y\}= \frac{3 \, e^{\left(-3 \, s\right)}}{s^{2} + 3 \, s + 2} + \frac{3}{s^{2} + 3 \, s + 2} </me>
    <me>
          \mathcal{L}\{y\}= -\frac{3 \, e^{\left(-3 \, s\right)}}{s + 2} + \frac{3 \, e^{\left(-3 \, s\right)}}{s + 1} - \frac{3}{s + 2} + \frac{3}{s + 1} </me>
    <me> {y} = 3 \, e^{\left(-t + 3\right)} \mathrm{u}\left(t - 3\right) - 3 \, e^{\left(-2 \, t + 6\right)} \mathrm{u}\left(t - 3\right) + 3 \, e^{\left(-t\right)} - 3 \, e^{\left(-2 \, t\right)} </me>
  </answer>
</exercise>