0008.tex

\begin{exerciseStatement}


 Explain how to solve the following IVP. 

\[ -8 \, \delta\left(t - 3\right) = -2 \, {y''} - 24 \, {y} + 14 \, {y'} \hspace{2em}
          y(0)= 0 ,
          y'(0)= 1 \]

Hint: \( \frac{1}{s^{2} - 7 \, s + 12} = -\frac{1}{s - 3} + \frac{1}{s - 4} \).


\end{exerciseStatement}
    
\begin{exerciseAnswer} 
\[
          \mathcal{L}\{y\}= \frac{4 \, e^{\left(-3 \, s\right)}}{s^{2} - 7 \, s + 12} + \frac{1}{s^{2} - 7 \, s + 12} \]\[
          \mathcal{L}\{y\}= -\frac{4 \, e^{\left(-3 \, s\right)}}{s - 3} + \frac{4 \, e^{\left(-3 \, s\right)}}{s - 4} - \frac{1}{s - 3} + \frac{1}{s - 4} \]\[ {y} = 4 \, e^{\left(4 \, t - 12\right)} \mathrm{u}\left(t - 3\right) - 4 \, e^{\left(3 \, t - 9\right)} \mathrm{u}\left(t - 3\right) + e^{\left(4 \, t\right)} - e^{\left(3 \, t\right)} \]
\end{exerciseAnswer}