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\begin{document}

\title{Applied Differential Equations and Modeling}
\author{Homework 11}
\date{Due in class Tuesday, May 7, 2019}
\maketitle

\begin{enumerate}

\item Find the inverse Laplace transforms of the given function.

\begin{enumerate}
\item $\displaystyle \frac3{s^2+4}$
\item $\displaystyle \frac{2s+1}{s^2-2s+2}$
\item $\displaystyle \frac{s^3-2 \, s^2 - 6 \,s - 6}{(s^2+2 \,s+2) \, s^2}$
\end{enumerate}


\item Solve the given initial value problem using the Laplace
transform. 

\begin{enumerate}
\item $\displaystyle y'' - y' - 6y = 0$

with $y(0)=1$, $y'(0)=-1$.

\emph{Note:} on the last homework set, you should have found that
\[
  Y(s) = \frac{s-2}{s^2-s-6} \,.
\]

\item $\displaystyle y'' + \omega^2 \, y = \cos 2t$

for $\omega^2 \neq 4$ with $y(0)=1$, $y'(0)=0$.

\emph{Note:} on the last homework set, you should have found that
\[
  Y(s) = \frac{s^3+5\,s}{(s^2+4)(s^2+\omega^2)} \,.
\]



\item $\displaystyle y''''- 4y = 0$

with $y(0)=1$, $y'(0)=0$, $y''(0)=-2$, $y'''(0)=0$.

\emph{Note:} on the last homework set, you should have found that
\[
  Y(s) = \frac{s^3}{s^4-4} \,.
\]




\end{enumerate}




\end{enumerate}
\end{document}
