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{\bf Name:}  \vrule height .4pt depth 0 pt width 2.0  
in  \hskip .5 in {\bf Instructor:} \vrule height .4pt depth 0 pt width 1.0 in  
  
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 \centerline{\bf {Mathematics 325}}  
\centerline{\bf Fall Semester 2000}  
\centerline{\bf Test 3, Version A}  
\centerline{\bf {November 16, 2000}}  
  
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\noindent  
1. This Examination contains twelve problems worth a total of  100  points.  
The test will be exactly  
75 minutes in length.  
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2. For each multiple choice question, please mark an $X$ on the correct  
answer on the answer sheet.  
{\bf Do not circle it.}  
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3. On problem 10, 11 and 12, show your work,  indicating clearly in the  
space provided how you arrive at your  
answer; otherwise no credit will be given.  
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4.  Calculators, books and notes are  
not allowed.  
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5. Hand in the entire test.  
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{\bf Sign the pledge:}  \lq \lq {On my honor,  
I have neither given nor received unauthorized aid on this Exam}":  
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\centerline{\bf GOOD LUCK}  
  
  
  
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\noindent  
{\bf PART A} (5 points each)  
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1.   
For the system 
$$ \frac{d {\bold x}}{d t} = \left( \matrix 
				0 & 5 \\ 
				3 & 0 \endmatrix \right){\bold x}, $$ 
the critical point $(0, 0)$  is an 
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a. asymptotically stable node. 
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b. unstable node. 
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c. asymptotically stable spiral point. 
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d. unstable spiral point. 
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e. unstable saddle point.  
 
 
  
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2.  
For the system 
$$ \frac{d {\bold x}}{d t} = \left( \matrix 
				0 & -5 \\ 
				3 & 0 \endmatrix \right){\bold x}, $$ 
the critical point $(0, 0)$  is 
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a. a center.  
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b. an asymptotically stable spiral point. 
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c. an unstable spiral point. 
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d. an asymptotically stable node. 
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e. an unstable node. 
 
  
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3.  
For the system 
$$ \frac{d {\bold x}}{d t} = \left( \matrix 
				1 & -1 \\ 
				0 & 1 \endmatrix \right){\bold x} 
				+ \left( \matrix 
					1 \\ 
					1 \endmatrix \right) 
, $$ 
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a. $(-2, -1)$ is a critical point and  is an unstable proper node.  
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b. $(-2, -1)$ is a critical point and is an unstable improper node.  
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c. $(1, 1)$ is a critical point and is an unstable improper node.  
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d. $(0, 0)$ is a critical point and is an unstable improper node.  
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e. $(0, 0)$ is a critical point and  is an unstable proper node.  
 
  
  
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\noindent  
4. For the non-linear system 
$$ \matrix \frac{d x}{d t} =  -x-y +  x^{3}y - x^{2} \\ 
   \hskip 1pt \frac{d y}{d t} = x-y +  xy - x^{2} y^{3}, \endmatrix $$ 
 
the critical point $(0, 0)$  is 
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a. a center. 
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b. an asymptotically stable node. 
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c. an unstable node. 
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d. an asymptotically stable spiral point.  
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e. an unstable spiral point. 
 
 
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5.    
Consider the almost linear system 
$$ \matrix \frac{d x}{d t} =  -x-y +  x^{2}y - y^{2} \\ 
   \hskip 7pt \frac{d y}{d t} = 3x+y +  5xy - x y^{2}. \endmatrix $$ 
 
The critical point $(0, 0)$  is 
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a. stable for both the corresponding linear system and the non-linear 
system. 
 
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b. asymtotically stable for both the corresponding linear system and  
the non-linear system. 
 
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c. stable for the corresponding linear system and asmptotically stable 
for the non-linear system. 
 
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d. stable for the corresponding linear system, but 
the stability for the non-linear system cannot be determined 
from the corresponding linear system. 
 
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e. stable for the corresponding linear system but not stable 
for the non-linear system. 
 
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6. 
Consider the critical point $(0, 0)$  for the almost linear system 
$$ \matrix \frac{d x}{d t} =  x-y +  x^{2} - y^{2} \\ 
   \hskip 15pt \frac{d y}{d t} = x+3y +  xy - x^{2} y. \endmatrix $$ 
 
 
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a. The critical point is  a spiral point. 
 
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b. The critical point is  an improper node. 
 
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c. The critical point is  a node 
 
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d. The stability of the critical point cannot be determined 
from the corresponding linear system. 
  
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e. The type of the critical point cannot be determined 
from the corresponding linear system. 
  
  
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7. 
Consider the non-linear system 
$$ \matrix \frac{d x}{d t} =  2x-xy  \\ 
   \hskip 2pt \frac{d y}{d t} = 3y+xy. \endmatrix $$ 
 
The critical point $(-3, 2)$ is a 
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a. saddle point 
 
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b. aymptotically stable node. 
 
 
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c. unstable node. 
 
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d. unstable spiral point 
 
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e. The stability of the critical point cannot be determined 
from the corresponding linear system. 
 
 
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8. Except for those on the $x$-axis and $y$-axis,  
the trajectories of the non-linear system  
$$ \matrix \frac{d x}{d t} =  x(1 + 2y)  \\ 
   \hskip 15pt \frac{d y}{d t} = y(-2 + 5x), \endmatrix $$ 
are described by the equation 
 
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a. $ -2y + 3xy  - x = C,$ where $C$ is a constant. 
 
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b. $\ln y -5y = \ln x + 2x + C, $ where $C$ is a constant. 
 
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c. $y = \frac{C}{x^{2}} e^{5x}, $ where $C$ is a constant. 
 
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d. $\ln y + 2y = -2 \ln x + 5x + C,$ where $C$ is a constant. 
 
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e. $y + y^{2} = -2x + \frac{5}{2} x^{2} + C,$ where $C$ is a constant. 
 
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9.  
The system 
$$ \matrix \frac{d x}{d t} =  8x^{3} - 4xy^{2}  \\ 
   \hskip 2pt \frac{d y}{d t} = -x^{2}y - y^{3} \endmatrix $$ 
has $(0, 0)$ as a critical point. 
Using $V(x, y) = x^{2} - 4 y^{2}$ as a Liapunov function 
to test the stability of $(0, 0)$, 
what conclusion can we get? 
 
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a. $(0, 0)$ is an asymptotically stable critical point. 
 
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b. $(0, 0)$ is a stable critical point. 
 
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c. $(0, 0)$ is an unstable critical point. 
 
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d. $(0, 0)$ is a stable but not asymptotically stable critical point. 
 
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e. The stability of $(0, 0)$ cannot be determined by Liapunov's method. 
 
 
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{\bf PART B} (15 points each)  
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10. Find the approximate value of the solution of the initial 
value problem 
$$ y'= 2y + 2t, \quad y(0) = 1 $$ 
at $t = 2$ using the Euler method with step size $h = 0.5$. 
 
  
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11. Consider the non-linear system: \hskip 20 pt 
$  \frac{d x}{d t} =  y-1,  
   \hskip 18pt \frac{d y}{d t} = 2e^{- x}  - 2y. $ 
 
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a. Find the critical point of this system. (It happens 
that there is only one critical point for this system.) 
 
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b. Write down the corresponding linear system for the critical point. 
Make the transformation $u = x - x_{0}$ and $v = y - y_{0}$ 
to move the critical point to the origin of the $uv$-plane, 
where $(x_{0}, y_{0})$ is a critical point of the original system. 
 
 
  
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c. Sketch a typical trajectory on the $uv$-plane for the corresponding 
{\bf linear} system and indicate the direction of motion for increasing $t$. 
 
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d. Discuss the type and stability of the critical point for the 
non-linear system. 
  
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12.   
Using Liapunov function of the form $V(x, y) = ax^{2} + c y^{2}$  
to test the stability of the critical point $(0, 0)$ of the 
system 
$$ \matrix \frac{d x}{d t} =  -4 x^{3} - xy^{2}  \\ 
   \hskip 2pt \frac{d y}{d t} =  8 x^{2}y - y^{3}. \endmatrix $$ 
You must state the correct part of the Liapunov's theorem 
and verify its conditions. 
 
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