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\begin{document}







\title{{\large\bf Math 325, Test 3}}
\comment{ \large 
\begin{minipage}[c]{6.2in}
\begin{center}{\bf READ THESE DIRECTIONS}\end{center}
\vspace*{1cm}
1. Please cross \fbox{$\times $} the correct answers. \\
2.  This test will be exactly 50 minutes in length.  
When you are told to begin, but not before, glance 
through the entire test and put your name on each
page.  It is YOUR RESPONSIBILITY to make sure your 
test consists of 5 PAGES with 7 PROBLEMS.  The point value for each 
multiple choice problem is 12 points. 
Problem 7 is worth 24 points (with additional 4 points for free).
Use the back of the test pages  for scratch work.\\
3.   On all partial credit problems, show your work, indicating clearly 
 how you arrived at your answer. 
 The points you receive for problem  7  will depend on the
extent to which the work you show convinces the grader 
(at the time of grading) that you did all or
part of the problem correctly.  
\end{minipage}} 
\answersheetfootline{\large\bf 
\fbox{\rule[-2mm]{0mm}{8mm}Sign your name:\hspace{6cm}}
\hspace{6cm}}
\twocolumn{12} 
\date{{\large December 2, 1992}}

\lastversion=3 

\noprofessor \nosection \vertmultchoice=2.5in
\multiplechoiceskip=0.8cm \let\ansborder=\boxborder \boxwidth=1cm
\boxdepth=0.4cm

\pagenumberstyle=1
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\problem
 Which  of the sketches on the last page of this exam
 best describes the trajectories of $x' = Ax$,
 $$ A = \left( \hspace{-2mm}
\begin{array}{cc} 5 & 1\\ 3 & 3\\  \end{array} \hspace{-2mm} \right)$$

\correct 4
\wrong 2
\wrong 3
\wrong 7
\wrong 1
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\problem
 Which  of the sketches on the last page of this exam
 best describes the trajectories of $x' = Ax$,
 $$ A = \left( \hspace{-2mm}
\begin{array}{cc} 1 & -1\\ 3 & 2\\  \end{array} \hspace{-2mm} \right)$$

\correct 8
\wrong 1
\wrong 7
\wrong 5
\wrong 2


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\problem
Consider the system
$$\begin{array}{clcr}
 x' & = & \frac{1}{y - 1} 
\\
\\
y' & = &x^{2} \end{array}.$$
Determine the trajectories of this
system by finding  $\frac{d y}{d x}$ and integrating.

\correct $y = C \exp \left( \frac{x^{3}}{3} \right) + 1$
\wrong $y = C \exp \left( \frac{x^{3}}{3} \right) - 1$
\wrong $y = C \left( \frac{x^{2}}{2} \right) + 1$
\wrong $y = \exp \left( \frac{x^{3}}{2} \right) - 1$
\wrong $x^{2} + y^{2} = C.$		

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\problem 
For the system
$$\begin{array}{clcr} x' & = & x + y \\ 
y' & = & -x - xy \end{array}$$
the critical point $(1,-1)$ is:

\correct a saddle point
\wrong a center
\wrong a spiral point. 
\wrong a proper node.
\wrong an improper node.

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\problem 
Consider the system:
$$\begin{array}{clcr} x' & = & - x^{3} \\ 
y' & = & - y^{3} .
\end{array}$$
Using a Liapunov function V(x,y) = $ax^{2} + cy^{2}$ one can say that
the critical point (0,0) is:

\correct asymptotically stable.
\wrong a saddle point.
\wrong a spiral point. 
\wrong stable but not asymptotically stable.
\wrong unstable.

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\problem 
(In the answers below  $n$ stands for all positive
integers and zero.)
\vspace{1mm}

\noindent
The critical points of the system
$$\begin{array}{clcr} x' & = & x \sin y \\ 
\\
y' & = & (x^{2} - 1)(y - 1) \end{array}$$
are:

\correct $(0,1),\; (1, \pm n \pi)$ and $(-1, \pm n \pi)$.
\wrong (0,1) and (1,0).
\wrong $(0, n \pi)$ and $(1, n \pi)$. 
\wrong $(n \pi, n \pi)$.
\wrong $(0,1), (1,0)$ and $(n \pi, n \pi)$.

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\problem
a) The equation
$$\frac{d^{2} \theta}{d t^{2}} + \sin \theta + 
\frac{d \theta}{d t} = 0$$
can be written
as a system of two first order differential equations
$$\begin{array}{clcr} x' & = &  y \\ 
\\
y' & = & - \sin x - y \end{array}$$
where $x=\theta $ and $y=\frac{d\theta }{d t}$.
\vspace{5mm}

\noindent
a)  List all critical points of this system.
\vspace{5mm}

\noindent
b)  Study the stability properties of the critical point $(0,0).$
\vspace{5mm}

\noindent
c) Study the stability properties of the critical point $(\pi ,0).$

\vspace{10cm}

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