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\begin{document}

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

\textbf{Exam III}\quad \textsl{Dec.~3, 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 (12 points)
Find the fundamental matrix $\Phi(t) = \exp(\vct{A}t)$, with $\Phi(0) =
\vct{I}$, for the homogeneous linear system $\vct{X}' = \vct{A}\cdot\vct{X}$
where
\[ \vct{A} = \left[ \begin{array}{rr}
	     1 & 2 \\ 2 & 1 \end{array}\right]
\]


\problemspace
\item (12 points)
Let
\[
\vct{A} = \left[ \begin{array}{rr}
         0 & 1 \\
         1 & 0 \end{array}\right]
\]
The general solution to the homogeneous system $\vct{X}' =
\vct{A}\cdot\vct{X}$ is
\[
\vct{X}(t) = c_1 \begin{vctr}1 \\ -1 \end{vctr} e^{-t} +
             c_2 \begin{vctr}1 \\  1 \end{vctr} e^{t}
\]
Using the method of variation of parameters, find a particular solution
of the nonhomogeneous system
\[
\vct{X}' = \vct{A}\cdot\vct{X} + \begin{vctr} e^t \\ e^{-t} \end{vctr}
\]


\problemspace
\item (12 points)
Consider the nonlinear system:
\begin{eqnarray*}
\frac{dx}{dt} &=& x + 2\sin(x) + 2y \\
\frac{dy}{dt} &=& 2x + 2y + e^y - 1
\end{eqnarray*}

  \begin{enumerate}[a)]
  \item Find the coefficient matrix for the corresponding linear system at
  the critical point $(0,0)$.
  \answerspace

  \item Find the eigenvalues and eigenvectors for this linear system.
  \answerspace

\newpage
  \item Classify the type and stability of the critical point $(0,0)$ for
  the nonlinear system.
  \answerspace

  \item Sketch several trajectories in the phase plane around $(0,0)$ for
  the nonlinear system.
  \answerspace

  \end{enumerate}


\problemspace
\item (5 points) Determine which of the following systems are autonomous.

\bigskip
  \begin{enumerate}[a)]
  \item $\begin{array}{rcl}
  \frac{dx}{dt} &=& \sin(y) + 3t \\[10pt]
  \frac{dy}{dt} &=& x + e^y
  \end{array}$

\bigskip
  \item $\begin{array}{rcl}
  \frac{dx}{dt} &=& e^y + 100x \\[10pt]
  \frac{dy}{dt} &=& \tan(x)
  \end{array}$

\bigskip
  \item $\begin{array}{rcl}
  \frac{dx}{dt} &=& 100x + 7y + e^t \\[10pt]
  \frac{dy}{dt} &=& -3x + 9y + \sin(t)
  \end{array}$

\bigskip
  \item $\begin{array}{rcl}
  \frac{dx}{dt} &=& y + t\\[10pt]
  \frac{dy}{dt} &=& 2 - x
  \end{array}$

\bigskip
  \item $\begin{array}{rcl}
  \frac{dx}{dt} &=& x^2 + y^2 \\[10pt]
  \frac{dy}{dt} &=& 1 - xy
  \end{array}$

  \end{enumerate}


\problemspace
\item (12 points)
Determine all real critical points of the following nonlinear system and
discuss their type and stability.
\begin{eqnarray*}
\frac{dx}{dt} &=& x - y \\
\frac{dy}{dt} &=& x - y^3
\end{eqnarray*}


\problemspace
\item (10 points)
The following nonlinear system models populations of competing species:
  \begin{eqnarray*}
  \frac{dx}{dt} &=& x(2 -  y - x)\\
  \frac{dy}{dt} &=& y(3 - 2x - y)
  \end{eqnarray*}
Determine which phase portrait corresponds to this system. Justify
your answer; simply choosing a phase portrait is not sufficient.

(a)$\begin{array}{l}\resizebox{2in}{!}{\includegraphics{vf1.eps}}\end{array}$
\hfill
(b)$\begin{array}{l}\resizebox{2in}{!}{\includegraphics{vf2.eps}}\end{array}$

(c)$\begin{array}{l}\resizebox{2in}{!}{\includegraphics{vf3.eps}}\end{array}$
\hfill
(d)$\begin{array}{l}\resizebox{2in}{!}{\includegraphics{vf4.eps}}\end{array}$

(e)$\begin{array}{l}\resizebox{2in}{!}{\includegraphics{vf5.eps}}\end{array}$

\problemspace
\item (12 points) The following nonlinear system models predator-prey
populations with logistic constraints that limit the growth of the prey
population, $x$, due to environmental factors:
  \begin{eqnarray*}
  \frac{dx}{dt} &=& x( 3 - x - y) \\
  \frac{dy}{dt} &=& y(-2 + x)
  \end{eqnarray*}
Determine the limiting populations $x(t)$, $y(t)$ as $t\to\infty$ if the
starting populations are $x(0) = 1$ and $y(0)=3$. Be sure to justify your
answer.

\problemspace
\item (10 points)
Use the method of separation of variables to replace the partial
differential equation
\[
(x^2 u_x)_x - u_{tt} = 0
\]
by a pair of ordinary differential equations.


%\problemspace
%\item (10 points)
%Find the solution of the heat conduction equation:
%\begin{eqnarray*}
%  u_{xx} &=& 9 u_t,\qquad 0 < x < 4,\quad t > 0 \\
%  u(0,t) &=& 0, \qquad u(4,t) = 0,\quad t > 0 \\
%  u(x,0) &=& \sin(\pi x/2) + 5\sin(\pi x),\qquad 0 \le x \le 4
%\end{eqnarray*}

\end{enumerate}

\end{document}
