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\begin{document}







 \centerline{\bf {Mathematics 325}}
\centerline{\bf Fall Semester 2002}
\centerline{\bf Test 3}
\centerline{\bf {November 21, 2002}}

\bigskip
\hrule
\bigskip
\noindent
1. This Examination contains twelve problems worth a total of  100  points.
(There are 10 problems in Part A worth a total of 60 points.
 There are 2 problems in Part B  worth a total of 30 points.  You begin
with 10 points).
The test will be exactly
75 minutes in length.
\bigskip
\noindent
2. For each multiple choice question, please mark an $X$ on the correct
answer on the answer sheet.
{\bf Do not circle it.}
\bigskip
\noindent
3. On problems   11 and 12, show your work,  indicating clearly in the
space provided how you arrive at your
answer; otherwise no credits will be given.
\bigskip
\noindent
4.  Calculators, books and notes are
not allowed.
\bigskip
\noindent
5. A table giving type and stability properties in terms of eigenvalues
is included at the end of the booklet (Table 9.3.1 from the text).
\bigskip
\noindent
6. Hand in the entire test.
\bigskip
\hrule
\bigskip
\bigskip



{\bf Sign the pledge:}  \lq \lq {On my honor,
I have neither given nor received unauthorized aid on this Exam}":
\bigskip
\hskip .7 in\vrule height .10pt depth 0 pt width 2 in
\bigskip
\bigskip
\centerline{\bf GOOD LUCK}




\newpage
 \noindent
{\bf PART A}
\vskip 2mm
\noindent
1. Consider the equation $$  y'=-4y+t^2, \ y(1)=2.$$ Use the Euler method
to find the approximate value of the solution
at $t=1.1$ with step size $h=0.1$.

\medskip
a. 1.2
\medskip
 b. 1.3
\medskip
 c.  1.4
\medskip
 d. 1.5
\medskip
 e. 1.6



\vskip 2cm
\noindent
 2. Consider the equation $$  y'=5y+t, \ y(0)=0.$$  Use the backward Euler
method to find the approximate value of the solution
at $t=0.1$ with step size $h=0.1$.

\medskip
 a. 0
\medskip
b.  0.1
\medskip
 c. 0.01
\medskip
 d. 0.02
\medskip
 e. 0.03


\newpage
\noindent
3.  Consider the system of equations
$$ \frac{d{\Bbb X}}{dt} = \left(\matrix    1 &1\\
 -1& 1\endmatrix \right) {\Bbb X}.$$
Classify the type and stability of the critical point at $(0,0)$.

\medskip
a. $(0,0)$ is a spiral point and is unstable.
\medskip
b. $(0,0)$ is a node and is asymptotically stable.
\medskip
c. $(0,0)$ is a saddle point and is unstable.
\medskip
d. $(0,0)$ is a center and is stable.
\medskip
e. $(0,0)$ is a spiral point and is asymptotically stable.

\vskip 2cm
\noindent
4. Find all critical points for the following system of nonlinear
differential equations:
$$\aligned \frac{dx}{dt}&=x(2-x-y)\\
           \frac{dy}{dt}&=y(-3+x+2y)\endaligned$$
\medskip
a. $(0,0)$
\medskip
b. $(0,0), (2,0) $
\medskip
c. $ (0,0), (2,0), (0,\frac 32) $
\medskip
d. $(0,0), (2,0), (0,\frac 32), (1,1)$
\medskip
e. $(0,0), (2,0), (3,0), (1,1)$





\newpage
\noindent
5. The system of nonlinear differential equations:

$$\aligned \frac{dx}{dt}&=-x-2y+x^2\\
           \frac{dy}{dt}&=-3+2x+y-xy\endaligned$$
\medskip
\noindent
has an isolated critical point at $(2,1)$. The linearization of this system
around the critical point $(2,1)$, using the coordinates $u=x-2$,
$v=y-1$, is

\medskip
a.  $$\frac{d}{dt} \left(\matrix u\\v \endmatrix \right)=
      \left(\matrix 3&2\\ 1&1\endmatrix \right)\left(\matrix u\\v \endmatrix
\right)$$
\medskip
b. $$\frac{d}{dt} \left(\matrix u\\v \endmatrix \right)=
      \left(\matrix 1&2\\ -1&-2\endmatrix \right)\left(\matrix u\\v
\endmatrix\right)$$
\medskip
c. $$\frac{d}{dt} \left(\matrix u\\v \endmatrix \right)=
      \left(\matrix 3&-2\\ 1&1\endmatrix \right)\left(\matrix u\\v \endmatrix
\right)$$
\medskip
d.  $$\frac{d}{dt} \left(\matrix u\\v \endmatrix \right)=
      \left(\matrix 3&-2\\ 1&-1\endmatrix \right)\left(\matrix u\\v \endmatrix
\right)$$
\medskip
e.  $$\frac{d}{dt} \left(\matrix u\\v \endmatrix \right)=
      \left(\matrix 2&3\\ 1&1\endmatrix \right)\left(\matrix u\\v \endmatrix
\right)$$

\bigskip
\noindent
6. Consider the nonlinear system of differential equations
$$\aligned \frac{dx}{dt}&=y\\
           \frac{dy}{dt}&=-5\sin x-2y \endaligned$$
Classify the type and stability of the critical point at $(0,0)$.

\medskip
a. $(0,0)$ is a spiral point and is unstable.
\medskip
b. $(0,0)$ is a node and is asymptotically stable.
\medskip
c. $(0,0)$ is a saddle point and is unstable.
\medskip
d. $(0,0)$ is a center and is stable.
\medskip
e. $(0,0)$ is a spiral point and is asymptotically stable.


\newpage
\noindent
7.  Consider the following phase plane diagram,
 with critical point at the origin.

\vskip 10cm

The critical point at the origin is
\medskip
a. An asymptotically stable node.
\medskip
b. An unstable node.
\medskip
c. An unstable saddle.
\medskip
d.  A stable spiral point.
\medskip
e. An unstable spiral point.

\newpage


8. Consider the linear differential system
$$\aligned \frac{dx}{dt}&=-2y\\
           \frac{dy}{dt}&= x\endaligned$$
Which of the following is a solution and solution trajectory for this system
with initial condition $x(0)=1$, $y(0)=0$. WARNING: you must match
the equation and the trajectory. The same equation can correspond to
several
different trajectories.\medskip
a. $2x^2+y^2=1$,  traveling clockwise
\medskip

b. $2x^2+ y^2=1$,  traveling counterclockwise
\medskip
c. $x^2+2y^2=1$,   traveling clockwise
\medskip
d. $ x^2+2y^2=1$,  traveling counterclockwise
\medskip
e. $x^2+4y^2=1$,   traveling clockwise




\bigskip
\noindent
9. The predator-prey system
$$\aligned \frac{dx}{dt}&=2x-xy=x(2-y)\\
           \frac{dy}{dt}&=-5y+xy=y(x-5)\endaligned$$
has critical points $(0,0)$ and $(5,2)$. Then


\medskip
a. $(0,0)$ is a node point and stable.
\medskip
b. $(0,0)$ is a node point and unstable.
 \medskip
c. $(0,0)$ is a saddle point and unstable.
\medskip
d. $(0,0)$ is a center and stable.
\medskip
e. $(0,0)$ is a spiral point and unstable.

\newpage
\noindent
10. Consider the oscillating pendulum without damping
$$\aligned \frac{dx}{dt}&=y\\
           \frac{dy}{dt}&=-\omega^2\sin x, \endaligned$$
where $\omega$ is a positive constant.
Classify the type and stability of the critical point at $(0,0)$.

\medskip
a. $(0,0)$ is a spiral point and is unstable.
\medskip
b. $(0,0)$ is a node and is asymptotically stable.
\medskip
c. $(0,0)$ is a saddle point and is unstable.
\medskip
d. $(0,0)$ is a center and is stable.
\medskip
e. $(0,0)$ is a spiral point and is asymptotically stable.

\newpage
\noindent
{\bf PART B}


\noindent
11. Consider the system
$$\cases dx/dt&= -2x+xy\\
          dy/dt&=-y+y^2 \endcases$$
\noindent (a) Find all  critical points.
\vskip 3cm

\noindent (b) At each critical point,  find the corresponding
linear  system and its eigenvalues.
\vskip 6cm
\noindent (c) At each critical point,  determine from (b) the type and
stability of the  nonlinear system.



\newpage

\noindent
12. Consider the nonlinear system of differential equations,
$$\cases  {dx}/{dt}&=-y-x^5\\
           {dy}/{dt}&=2x-y^5\endcases$$
with the critical point $(0,0)$.
\vskip 5mm
\noindent (a) For the linearization of this system near $(0,0)$,
determine the type and stability for this linear system.
\vskip 4cm
\noindent (b) Using a Liapunov function of the form $V(x,y)=ax^2+cy^2$,
determine the type and stability of the critical point
$(0,0)$ for the above nonlinear
system. State the Liapunov function you are using.

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