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\begin{document}

\textbf{Math 325: Differential Equations}
\hfill\makebox[2.5in][r]{Name:\hrulefill}

\textbf{Exam II}\quad \textsl{Oct.~31, 1996}
\hfill\makebox[2.5in][r]{Section:\hrulefill}

\bigskip
There are 8 problems worth total of 85 points. You start with 15 points.

\bigskip

\begin{enumerate}
\item (10 points)

  \begin{enumerate}[a)]
  \item Use the Laplace transform to solve the initial value problem
  \[
  y'' - 2 y' + 10 y = \delta(t),\quad y(0) = y'(0) = 0
  \]
  where $\delta(t)$ is the Dirac delta function.
  \answerspace

  \item Let $f(t)$ be an unspecified function that has a Laplace
  transform. Express the solution to the following initial value
  problem as an integral.
  \[
  y'' - 2 y' + 10 y = f(t),\quad y(0) = y'(0) = 0
  \]
  \answerspace

  \end{enumerate}

\problemspace
\item (15 points) Solve the following initial value problem using Laplace
transforms.
\[
y'' + 3y' + 2y = g(t), \quad y(0) = 0, y'(0) = 1
\]
where
\[
g(t) = \cases{1 & $0\le t \le 1$\cr
              0 & $t \ge 1$\cr}
\]

\problemspace
\item (10 points)
  Let $\vct{X}^{(1)}(t) = \begin{vctr} t \\ t \\ t^2 \end{vctr}$,
  $\vct{X}^{(2)}(t) = \begin{vctr} t^2 \\ t \\ t \end{vctr}$,
  $\vct{X}^{(3)}(t) = \begin{vctr} t \\ t^2 \\ t \end{vctr}$,

  \begin{enumerate}[a)]
  \item
  Compute the Wronskian of
  $\vct{X}^{(1)}(t)$, $\vct{X}^{(2)}(t)$, $\vct{X}^{(3)}(t)$.
  \answerspace
  \answerspace

  \item 
  For what values of $t$ are the vectors 
  $\vct{X}^{(1)}(t)$, $\vct{X}^{(2)}(t)$, $\vct{X}^{(3)}(t)$ linearly
  dependent.
  \answerspace

  \item
  On what intervals are the vector functions
  $\vct{X}^{(1)}(t)$, $\vct{X}^{(2)}(t)$, $\vct{X}^{(3)}(t)$ linearly
  independent.
  \answerspace

  \end{enumerate}

\problemspace
\item (10 points) Classify the following systems of first order
equations using as many of the following terms as apply: linear,
non-linear, homogeneous, non-homogeneous. (Do not try to solve the
equations!)

  \begin{enumerate}[a)]
  \item
  \begin{eqnarray*}
  x_1' &=& t x_1  + (t^2 - e^t)x_2  \\
  x_2' &=& -(t^3 - e^{-t})x_1 + \frac{1}{t}x_2 \\
  \end{eqnarray*} \answerspace

  \item
  \begin{eqnarray*}
  x_1' &=& \cos(t)x_2 - \sin(t) \\
  x_2' &=& \sin(t)x_1 + \cos(t) \\
  \end{eqnarray*}
  \answerspace

  \item
  \begin{eqnarray*}
  x_1' &=& 1/x_2 \\
  x_2' &=& 1/x_1 \\
  \end{eqnarray*}
  \answerspace

  \end{enumerate}

\problemspace
\item (10 points) Find the eigenvalues and corresponding eigenvectors
of the matrix
\[
  \vct{A} =
  \left[\begin{array}{rrr}
  4 & 0 & 0 \\
 -1 & 2 &-3 \\
  2 & 4 &10 \\
  \end{array}\right]
\]

\problemspace
\item (10 points) Find the general solution of the system
\[
\vct{X}' = \left[\begin{array}{rrr}
           1 & 2 & 1 \\
           0 & 5 & 3 \\
           0 & 3 & 5 \\ \end{array}\right] \vct{X}
\]

\problemspace
\item (10 points) Find the solution of the initial value problem
\[
\vct{X}' = \left[\begin{array}{rr}
         4 & -7 \\
         7 &  4 \\ \end{array}\right] \vct{X},\quad
         \vct{X}(0) = \begin{vctr}2 \\ 3\end{vctr}
\]
The coefficient matrix has a complex eigenvlaue of $4+7i$ and a
corresponding complex eigenvector $\begin{vctr}i\\ 1\end{vctr}$.

\problemspace
\item (10 points) Find the general solution of the system
\[
\vct{X}' = \left[\begin{array}{rr}
         -6 &  4 \\
         -1 & -2 \\ \end{array}\right] \vct{X}
\]

\end{enumerate}

\end{document}
