0009.tex

\begin{exerciseStatement}


 Explain how to solve the following IVP. 

\[ 3 \, {y''} = -27 \, {y} - 81 \, \mathrm{u}\left(t - 2\right) \hspace{2em}
          y(0)= -2 ,
          y'(0)= 0 \]

Hint: \( \frac{1}{s^{3} + 9 \, s} = -\frac{s}{9 \, {\left(s^{2} + 9\right)}} + \frac{1}{9 \, s} \).


\end{exerciseStatement}
    
\begin{exerciseAnswer} 
\[
          \mathcal{L}\{y\}= -\frac{2 \, s}{s^{2} + 9} - \frac{27 \, e^{\left(-2 \, s\right)}}{{\left(s^{2} + 9\right)} s} \]\[
          \mathcal{L}\{y\}= \frac{3 \, s e^{\left(-2 \, s\right)}}{s^{2} + 9} - \frac{2 \, s}{s^{2} + 9} - \frac{3 \, e^{\left(-2 \, s\right)}}{s} \]\[ {y} = 3 \, \cos\left(3 \, t - 6\right) \mathrm{u}\left(t - 2\right) - 2 \, \cos\left(3 \, t\right) - 3 \, \mathrm{u}\left(t - 2\right) \]
\end{exerciseAnswer}