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\begin{document}

\title{Applied Differential Equations and Modeling}
\author{Homework 8}
\date{Due in class Tuesday, May 15}
\maketitle

\begin{enumerate}

\item Find the solution of the given initial value problem and draw
its graph.
\begin{enumerate}
\item $\displaystyle y'' +2 y' + 2y = \delta(t-\pi)$
with $y(0)=1$, $y'(0)=0$

\item $\displaystyle y'' + y = \delta(t-2 \pi) \, \cos t$
with $y(0)=0$, $y'(0)=1$

\end{enumerate}

\item Find the Laplace transform of the given function.

\begin{enumerate}
\item $\displaystyle \int_0^t (t-\tau)^2 \, \cos 2\tau \, \d \tau$

\item $\displaystyle \int_0^t \e^{-(t-\tau)} \, \sin\tau \, \d \tau$


\end{enumerate}

\item Find the inverse Laplace transform using the convolution theorem.

\begin{enumerate}
\item $\displaystyle F(s) = \frac1{s^4 \, (s^2+1)}$

\item $\displaystyle F(s) = \frac{G(s)}{s^2+1}$


\end{enumerate}


\item In each of the following problems, express the total response in
terms of the forced response (using a convolution integral) and the
free response.
\begin{enumerate}
\item $\displaystyle y'' + \omega^2 \, y = g(t)$
with $y(0)=0$, $y'(0)=1$

\item $\displaystyle y'' + 3 y' + 2 y = \cos (\alpha t)$
with $y(0)=1$, $y'(0)=0$

\end{enumerate}

\item Which of the following transfer functions corresponds to a
BIBO-stable system?

\begin{enumerate}
\item $\displaystyle H(s) = \frac1{(s+1)^2 \, (s^2+1)}$

\item $\displaystyle H(s) = \frac1{s^2-1}$

\item $\displaystyle H(s) = \frac1{6+11s+6s^2+s^3}$


\end{enumerate}



\end{enumerate}
\end{document}
