\magnification=\magstep1
\input epsf
\def\everymath{\displaystyle}
\def\problemspace{0.5truein}

\def\bmatrix#1{\left[\matrix{#1}\right]}
\def\X{{\bf X}}
\def\Y{{\bf Y}}
\def\U{{\bf U}}
\def\A{{\bf A}}
\def\G{{\bf G}}
\def\P{{\bf P}}
\def\0{{\bf 0}}


%% TITLE
\noindent
{\bf Math 325: Differential Equations}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam III}\quad{\sl November 25, 1997}\hfill
Section:\hbox to 2.5truein{\hrulefill}



 0.5truein\noindent
Record your answers to the multiple choice problems by placing an
$\times$ through one letter for each problem on this answer sheet.
There are 10 multiple choice questions worth 5 points each and three
partial credit problems worth 10 points each. You start with 20~points.

\def\answerspace{0.5in}

%% PROBLEM
Find a fundamental matrix for the system
$\X' = \bmatrix{ 9 & -8\cr 12 & -11 \cr}\cdot \X$ 
%% CORRECT
$\bmatrix{2e^{-3t} & e^t\cr 3e^{-3t} & e^t}$
%% WRONG
$\bmatrix{e^{-3t} & e^t\cr e^{-3t} & e^t}$
%% WRONG
$\bmatrix{3e^{-3t} & 2e^t\cr e^{-3t} & -e^t}$
%% WRONG
$\bmatrix{2e^{t} & e^{-3t}\cr e^{t} & 3e^{-3t}}$
%% WRONG
$\bmatrix{-e^{t} & e^{-3t}\cr e^{t} & e^{-3t}}$

%% PROBLEM
Suppose
  $$\Psi(t) = \bmatrix{2\cos(t)-\sin(t) & \cos(t) + 2\sin(t)\cr
  \cos(t) & \sin(t)}$$
is a fundamental matrix for a system of linear first order equations.
Determine the\hfil\break unique solution through the point $(1,2)$.
%% CORRECT
$(\cos(t)-8\sin(t),2\cos(t)-3\sin(t))$
%% WRONG
$(\cos(t),2\cos(t))$
%% WRONG
$(\cos(t)-4\sin(t),2\cos(t)+2\sin(t))$
%% WRONG
$(\cos(t)+3\sin(t),2\cos(t)+8\sin(t))$
%% WRONG
$(\cos(t)+2\sin(t),2\cos(t)-4\sin(t))$

%% PROBLEM
Let $\Psi(t)$ be a fundamental matrix for a homogeneous linear system
$\X' = \P(t)\cdot \X$.\hfil\break
A solution to the non-homogeneous system $\X' = \P(t)\cdot\X + \G(t)$ can be
written in the form
$\Y(t) = \Psi(t)\cdot\U(t)$ where $\U(t)$ satisfies:
%% CORRECT
$\U'(t) = \Psi(t)^{-1}\cdot \G(t)$
%% WRONG 
$\U'(t) = \Psi(t)\cdot \G(t)$
%% WRONG
$\U'(t) = \Psi(t)\cdot\Psi(0)^{-1}\cdot \G(t)$
%% WRONG
$\U'(t) = \int \Psi(t)\cdot\Psi(0)^{-1}\cdot \G(t)$
%% WRONG
$\U'(t) = \int \Psi(t)^{-1}\cdot \G(t)$

%% PROBLEM
Let $A = \bmatrix{0 & 1\cr 1 & 0}$. Which of the following matrices
equals $\exp(A t)$.
%% CORRECT
$\bmatrix{\cosh(t) & \sinh(t) \cr \sinh(t) & \cosh(t)}$
%% WRONG
$\bmatrix{\cos(t) & -\sin(t) \cr -\sin(t) & \cos(t)}$
%% WRONG
$\bmatrix{e^t & e^{-t}\cr e^t & -e^{-t}}$
%% WRONG
${1\over2}\bmatrix{-e^t & e^{-t}\cr e^t & e^{-t}}$
%% WRONG
$\bmatrix{e^t & 0\cr 0 & e^{-t}}$

%% PROBLEM
Determine the type and stability of the critical point $(0,0)$ for the system
$$\eqalign{
{dx\over dt} &= 2x + 2y\cr
{dy\over dt} &= 2x - 2y\cr}$$
%% CORRECT
saddle
%% WRONG
unstable node
%% WRONG
unstable spiral
%% WRONG
stable spiral
%% WRONG
stable node

%% PROBLEM
Consider the non-linear system
$$\eqalign{
{dx\over dt} &= \cos(x) - y^2\cr
{dy\over dt} &= e^{xy} - 1\cr}$$
Find the coefficient matrix for the corresponding linear system at the
critical point\hfil\break$(0,1)$.
%% CORRECT
$\bmatrix{0&-2\cr1&0\cr}$
%% WRONG
$\bmatrix{-1&0\cr1&1\cr}$
%% WRONG
$\bmatrix{-2&0\cr0&1\cr}$
%% WRONG
$\bmatrix{-3&0\cr0&0\cr}$
%% WRONG
$\bmatrix{0&-1\cr1&1\cr}$

%% PROBLEM
Determine the critical points of the system
$$\eqalign{
{dx\over dt} &= x - x^2 - xy\cr
{dy\over dt} &= 2y - y^2 - 3xy\cr}$$
%% CORRECT
$(0,0)$, $(0,2)$, $(1,0)$, $(1/2,1/2)$
%% WRONG
$(0,0)$, $(0,2)$, $(1,0)$, $(1,1)$
%% WRONG
$(0,0)$, $(0,1)$, $(2,0)$, $(2,2)$
%% WRONG
$(0,0)$, $(0,1/2)$, $(1,0)$, $(1/2,1/2)$
%% WRONG
$(0,0)$, $(2,0)$, $(0,1)$, $(1,1)$

%% PROBLEM
Determine the type and stability of the critical point $(0,0)$ for the
system
$$\eqalign{
{dx\over dt} &= 2x + 4y^2\cr
{dy\over dt} &= x + 2y\cr}$$
%% CORRECT
unstable node or spiral
%% WRONG
asymptotically stable node
%% WRONG
center, stability indeterminate
%% WRONG
unstable saddle
%% WRONG
unstable spiral

%% PROBLEM
Identify which type of system most closely matches the following phase
portrait.
$$\epsfxsize=3truein\epsfbox{exam3.1.eps}$$
%% CORRECT
predator-prey system
%% WRONG
undamped pendulum
%% WRONG
damped pendulum
%% WRONG
competing species with stable coexistence
%% WRONG
competing species with  extinction of one species


%% PROBLEM
The following system models populations of competing species.
$$\eqalign{
{dx\over dt} &= x(2 - 2x - y)\cr
{dy\over dt} &= y(4 - 2x - 3y)\cr}$$
Deteremine the limiting populations $x(t)$, $y(t)$, as $t\to\infty$, if the
starting populations are $x(0) = 3$ and $y(0) = 4$.
%% CORRECT
$x(t)\to1/2$, $y(t)\to1$
%% WRONG
$x(t)\to1$, $y(t)\to0$
%% WRONG
$x(t)\to0$, $y(t)\to4/3$
%% WRONG
$x(t)\to0$, $y(t)\to0$
%% WRONG
$x(t)\to3/2$, $y(t)\to2$

%% END
\vfil\eject
\noindent
11.
Find the general solution of the following system using the method of
variation of parameters.
$$\eqalign{
{dx\over dt} &= -10x + 3y - 3e^{-t}\cr
{dy\over dt} &= -36x + 11y - 6e^{-t}\cr}$$

\vfil\eject
\noindent
12.
Determine all the real critical points of the following system and
discuss the type and stability of each.
$$\eqalign{
{dx\over dt} &= 1 - xy\cr
{dy\over dt} &= x - y^3\cr}$$


\vfil\eject
\noindent
13.
Consider the following system that models two interacting species with
populations $x$ and $y$.
$$\eqalign{
{dx\over dt} &= x(1 - {1\over2}y)\cr
{dy\over dt} &= y(-{1\over3}+{2\over3}x)\cr}$$

\noindent
{\it(a)} Find the critical points of this system and describe their
stability properties.


\noindent
{\it (b)} Sketch a phase portrait for this system.

\end
