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\title{Applied Differential Equations and Modeling}
\author{Homework 6}
\date{Due in class Tuesday, April 17}
\maketitle

\begin{enumerate}

\item Find the solution to the following initial value problems.
\begin{enumerate}
\item $\displaystyle
 y'' + 4\,y = t^2 + 3\,e^t \,, \quad
 y(0) = 0 \,, \quad
 y'(0) = 2
$

\item $\displaystyle 
  y'' - 2y' - 3y = 3t \, e^{2t} \,, \quad
  y(0) = 1 \,, \quad y'(0) = 0
$

\end{enumerate}

\item Find the general solution to the initial value problem 
\begin{equation*}
   u'' + \omega_0^2 \, u = \cos \omega t
\end{equation*}
for 
\begin{enumerate}
\item $\omega \neq \omega_0$,
\item $\omega = \omega_0$.
\end{enumerate}

\item Find the gain function $\lvert G(\i \omega) \rvert$  for the
vibrating system described by the initial value problem
\[
  y'' + 0.25 \, y' + 2 \, y = 2 \, \cos \omega t \,, \quad
  y(0) = 0 \,, \quad y'(0) = 2 \,.
\]
For which value of $\omega$ is the the gain maximal?  Is this value
smaller or larger than the resonance frequency of the undamped
equation? 



\item Consider a constant coefficient second order equation with
inhomogeneous right hand side, i.e.
\begin{equation*}
  a \, y'' + b \, y' + c \, y = g(t) \,. \tag{*}
\end{equation*}
Show that if the characteristic equation
\[
  a \, \lambda^2 + b\, \lambda + c = 0 
\]
has two roots with negative real part, then all solutions to the
differential equation coincide asymptotically.  In other words, if
$y_1$ and $y_2$ are two solutions of (*), then
\[
  \lim_{t \to \infty} (y_1(t) - y_2(t)) = 0 \,.
\]


\end{enumerate}
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