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\begin{document}

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

\textbf{Final Exam}\quad \textsl{Dec.~19, 1996}
\hfill\makebox[2.5in][r]{Section:\hrulefill}

\bigskip
There are 15 problems worth total of 150 points. Be sure to
justify your answers and show all your work.

\bigskip


\begin{enumerate}

\item (10 points)
Find the general solution of the nonhomogeneous differential equation
\[
y''' + y = x e^{-x}
\]

\problemspace
\item (10 points)
Use the Runge-Kutta method with step size $h = 0.1$ to find an
approximate value of the solution at $t=1.1$ of the initial value
problem
\[
y' = t + e^y,\qquad y(1) = 0
\]
Be sure to indicate carefully the formulas you are using and show every
step of the calculation to four decimal places. (If you cannot remember
the Runge-Kutta formulas, you can take a 3 point penalty and solve this
problem using the Euler method.)

\problemspace
\item (10 points)

  \begin{enumerate}[a)]
  \item Define the Laplace transform $\LL\{f(t)\}$ of a function $f(t)$.
  \answerspace

  \item Write $\LL\{f^{(n)}(t)\}$ in terms of $\LL\{f(t)\}$ and the initial
  values of $f(t)$ and its derivatives.
  \answerspace

  \end{enumerate}

\problemspace
\item (10 points)
 Use the Laplace transform to solve the initial value problem
\[
  y'' + y = \delta(t -\pi) - u_{2\pi}(t),\qquad y(0) = 1, \quad y'(0) = 0
\]
where $\delta(t)$ is the unit impulse function and $u_{2\pi}(t)$ is a unit
step function.

\problemspace
\item (10 points)

  \begin{enumerate}[a)]
  \item Define the convolution $f*g(t)$ of two functions $f(t)$ and $g(t)$.
  \answerspace

  \item Express $\LL^{-1}\left\{\frac{1}{s^4(s^2+1)}\right\}$ as an
  integral.  \answerspace

  \end{enumerate}

%\problemspace
%\item (10 points)
%Convert the differential equation
%\[
%y''' + x^2y'' - xy' + y = e^x
%\]
%into a system of first order differential equations.

\problemspace
\item (10 points)
Prove that the functions $e^x$, $e^{-x}$, $e^{2x}$ are linearly independent.

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

\problemspace
\item (10 points)
Find the general solution of the system
$\vct{X}' = \vct{A}\cdot\vct{X}$ where $\vct{A}$ is the matrix
\[
\vct{A} = \left[\begin{array}{rr}
1 & -5\\
2 & -1
\end{array}\right]
\]

\problemspace
\item (10 points)
Use the method of variation of parameters to find a particular solution
of the system
\[
\vct{X}' = \left[\begin{array}{rr}
-1 & 1\\
1 & -1\end{array}\right]\cdot\vct{X}
+ \begin{vctr}e^t \\ e^t\end{vctr}
\]

\problemspace
\item (10 points)
Find all critical points of the system
\begin{eqnarray*}
\frac{dx}{dt} &=& y^2 - 1\\
\frac{dy}{dt} &=& x^3 - y
\end{eqnarray*}
and determine their type and stability.


\problemspace
\item (10 points)
Consider the predator-prey system
\begin{eqnarray*}
\frac{dx}{dt} &=& x - xy\\
\frac{dy}{dt} &=& -y + xy
\end{eqnarray*}

  \begin{enumerate}[a)]
  \item Find the critical points of this system and describe
  their stability properties.
  \answerspace

  \item Sketch the phase portrait for this system.
  \answerspace

  \end{enumerate}

\problemspace
\item (10 points)
Solve the heat equation $3 u_{xx} = u_{t}$ subject to the homogeneous
boundary conditions
\[
u(0,t) = 0 = u(50,t),\qquad t > 0
\]
and initial condition
\[
u(x,0) = 100
\]

\problemspace
\item (10 points)
Solve the heat equation $\alpha^2 u_{xx} = u_{t}$ subject to the
non-homogeneous boundary conditions
\[
u(0,t) = 0,\quad  u(\ell,t) = 50,\qquad t > 0
\]
and initial condition
\[
u(x,0) = \frac{50}{\ell}x
\]

\problemspace
\item (10 points)
Solve the wave equation $4u_{xx} = u_{tt}$ subject to the homogeneous
boundary conditions
\[
u(0,t) = 0 = u(\pi,t)
\]
and initial conditions
\begin{eqnarray*}
u(x,0) &=& \frac{1}{10} \sin(2x)\\
u_t(x,0) &=& 0
\end{eqnarray*}

\problemspace
\item (10 points)
Find the Fourier series expansion for the function

\raisebox{1in}{
$f(x) = \left\{\begin{array}{ll}
x+2 & -2 \le x \le 0\\
2-x & 0 \le x \le 2
\end{array}\right.$}
\resizebox{3in}{!}{\includegraphics{final.eps}}

\end{enumerate}

\end{document}
