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\centerline{\textbf{\large TEST}}

\vspace*{.3in}

\textbf{MATxxx - Semester 200x  \hspace*{1.5in} NAME \hrulefill}

\vspace*{.1in}

\textbf{MWF ??:??-??:??   \hspace*{2.4in} I.D. \hrulefill}

\vspace*{.3in}

\noindent
Closed notes/book. Calculators OK. Show and explain your work.
Circle your answers. Staple extra sheets.

\vspace*{0.6in}

We consider the IVP
$\left\{
\begin{array}{l}
   \frac{1}{2}y''+y'+\frac{5}{2}y=e^{-t}\sin(2t) \\
   y(0)=1 \\
   y'(0)=-\frac{3}{2}
\end{array}
\right.$

\begin{enumerate}
\item
Determine the general solution of the HODE.

\item
Verify that the functions $y_1(t) = e^{-t}\cos(2t)$ and $y_2(t) =
e^{-t}\sin(2t)$ are linearly independent on $\mathbb{R}$.

\item
Determine the \emph{form} of a particular solution using the method
of undetermined coefficients (\textbf{you do not have to find the
coefficients}).

\item
Find a particular solution \underline{using the method of variation
of the constants}.

\item \label{q4}
Solve the IVP.

\item
Transform the IVP into a set of first order ODEs with appropriate
initial conditions.

\item
Find an approximation to $y(0.5)$ \underline{and} $y'(0.5)$ using two
steps of Euler's method.

\item   % EXAMPLE OF TABLE
The following table yields the numerical estimates of $y(0.5)$ using
$n = 4, 8, 16$ and $1000$ intervals, respectively:
\begin{table}[h]
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|} \hline
n & 2 & 4 & 8 & 16 & 32 & 64 & 1000 \\\hline
estimate of $y(0.5)$ & 0.125 & 0.173 & 0.209 & 0.217 & 0.237 & & 0.246 \\\hline
\end{tabular}
\caption{This is table 1}
\end{center}
\end{table}

One entry is wrong. Which one is it? Fill in the missing data in the
table.

\item
Determine the Laplace transform $Y(s)$ of the solution of the IVP by
applying the Laplace transform directly to the ODE (i.e., do
\underline{not} use the answer from Question \ref{q4}).

\item
Using the fact that ${\cal
L}^{-1}\left\{\frac{6}{(s^2+2s+5)^2}\right\} =
e^{-t}\left(\sin(2t)-2t\cos(2t)\right)$, find the solution of the IVP
from its Laplace transform. See Figure \ref{fig1}.
\begin{figure}[h]
\centering{\caption{this is a
triangle}\epsfig{figure=triangle.eps,height=2.0in,angle=-90}}
\label{fig1}
\end{figure}

\end{enumerate}

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