0004.tex

\begin{exerciseStatement}


 Explain how to solve the following IVP. 

\[ -12 \, {y} + 12 \, \delta\left(t - 1\right) = -3 \, {y''} \hspace{2em}
          y(0)= 0 ,
          y'(0)= -8 \]

Hint: \( \frac{1}{s^{2} - 4} = -\frac{1}{4 \, {\left(s + 2\right)}} + \frac{1}{4 \, {\left(s - 2\right)}} \).


\end{exerciseStatement}
    
\begin{exerciseAnswer} 
\[
          \mathcal{L}\{y\}= -\frac{4 \, e^{\left(-s\right)}}{s^{2} - 4} - \frac{8}{s^{2} - 4} \]\[
          \mathcal{L}\{y\}= \frac{e^{\left(-s\right)}}{s + 2} - \frac{e^{\left(-s\right)}}{s - 2} + \frac{2}{s + 2} - \frac{2}{s - 2} \]\[ {y} = -e^{\left(2 \, t - 2\right)} \mathrm{u}\left(t - 1\right) + e^{\left(-2 \, t + 2\right)} \mathrm{u}\left(t - 1\right) - 2 \, e^{\left(2 \, t\right)} + 2 \, e^{\left(-2 \, t\right)} \]
\end{exerciseAnswer}