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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 Final Exam.}
\centerline{\bf {December 13, 2000}}
\bigskip
\bigskip
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\hrule
\bigskip
\noindent
1. This Examination contains 21 problems worth a total of  150  points.
  The test will be exactly
120 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.  Calculators, books and notes are
not allowed.
\bigskip
\noindent
4. A table of Laplace transforms is included at the end of the booklet.
\bigskip
\noindent
5. 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}":
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\hskip .7 in\vrule height .10pt depth 0 pt width 2 in
\bigskip
\bigskip
\centerline{\bf HAVE A NICE VACATION}


\newpage


\bigskip 
\noindent 
1.  The Wronskian of $\{1, e^{2x}, e^{-2x} \}$ is

(a) $16$

(b) $2$

(c)  $e^{2x}$

(d) $e^{4x}$

(e)  $1$

\bigskip 
\noindent 
2. The general solution of  the equation $$y^{(3)} + 3y'' + 2y' = 0$$ is 

(a) $c_1+ c_2 e^{-t} + c_3 e^{-2t}$

(b) $c_1e^{3t} + c_2e^{2t} + c_3e^t$

(c) $c_1 + c_2 e^t + c_3 e^{2t}$

(d) $c_1 e^{-t}+ c_2 e^{-2t} + c_3 e^{-3t}$


(e) $c_1+ c_2 t + c_3 t^2$




\bigskip
\noindent
3. Determine a suitable form for a particular solution of the equation
$$y'''- 9 y' = t + e^{3t}.$$

(a) $t(A_0 t + A_1) + B t e^{3t}$

(b) $A_0 t + A_1 + B  e^{3t}$

(c) $t(A_0 t + A_1) + B  e^{3t}$

(d) $t(A_0 t + A_1) + B$

(e) $A_0 t  + B t e^{3t}$


\bigskip
\noindent
4. Given that $y_1(t) =1,$ $y_2(t) = t $ and $y_3(t) = e^t $ 
form a fundamental set of  solutions of the equation 
$y''' - y'' = 0$, determine a particular solution of 
$$ y''' - y'' = g(t)$$
in terms of an integral.

(a) $\int^t_{t_0} (s-1) g(s) ds - t \int^t_{t_0}  g(s) ds + e^t 
\int^t_{t_0} e^{-s} g(s) ds$

(b) $\int^t_{t_0} e^s(s-1) g(s) ds - t \int^t_{t_0} e^s g(s) ds + e^t 
\int^t_{t_0}  g(s) ds$


(c) $\int^t_{t_0} e^s(s-1) g(s) ds -  \int^t_{t_0} e^s g(s) ds + 
\int^t_{t_0}  g(s) ds$


(d) $\int^t_{t_0} (s-1) g(s) ds -  \int^t_{t_0}  g(s) ds +  
\int^t_{t_0} e^{-s} g(s) ds$

(e) $\int^t_{t_0} s g(s) ds - t \int^t_{t_0} e^s g(s) ds + e^t 
\int^t_{t_0} e^{-s} g(s) ds$

\newpage
\bigskip
\noindent
5. Let
$$ f(t)=
\left\{
\aligned  &100, \quad \quad 0< t\le 99 \\
                        & 0, \quad \quad t > 99. \endaligned
\right.
$$
Find the Laplace transform of $f$.
\medskip
(a)  $   \dsize\frac{100}{s}(1 - e^{-99s})$   
\medskip
(b) $   \dsize\frac{99}{s}(1 - e^{-100s})$
\medskip
(c) $   100(1 - e^{-99s})$
\medskip
(d)  $   99(1 - e^{-100s})$
\medskip
(e) $   \dsize\frac{100}{s}(1 - e^{99s})$



\bigskip
\noindent
6. The inverse Laplace transform of $$\frac{2s + 5 }{s^2+ 5s + 6}$$ is
\medskip
(a) $e^{-2t} + e^{-3t}$
\medskip
(b)  $e^{2t} + e^{3t}$
\medskip
(c)  $\sin {2t} + \sin {3t}$
\medskip
(d)  $e^{2t}\cos{2t} + e^{3t}\sin {3t}$
\medskip
(e)  $e^{3t}\cos{2t}$






\bigskip
\noindent
7. The solution of the initial value problem
$$y''+4y= \delta (t-\frac \pi 4),\quad y(0)=y'(0)=0 $$
is
\medskip
(a) $\frac 12 \sin 2(t-\frac \pi 4)$
\medskip
(b) $ \frac 12 \cos 2(t-\frac \pi 4)$
\medskip
(c) $\delta(t-\frac\pi 4) \sin 2(t-\frac \pi 4)$
\medskip
(d) $ \frac 12 u_{\frac\pi 4}(t)\sin 2(t-\frac \pi 4)$
\medskip
(e) $ \frac 12 u_{\frac\pi 4}(t)\cos 2(t-\frac \pi 4)$






\bigskip 
\noindent 
8.  Find the general solution of the system of equations
$$ \frac{d {\bold x}}{d t} = \left( \matrix
				2 & 0 & 0 \\
				0 & 1 & 1 \\
				0 & 0 & 1 \endmatrix \right){\bold x} $$

\medskip 
(a) $ c_{1} e^{2t} \left( \matrix 1 \\ 0 \\ 0 \endmatrix \right)
	+ c_{2} e^{t} \left( \matrix 0 \\ 1 \\ 0 \endmatrix \right)
	+ c_{3} (te^{t} \left( \matrix 0 \\ 1 \\ 0 \endmatrix \right)
	+ e^{t} \left( \matrix 0 \\ 0 \\ 1 \endmatrix \right))$
\medskip 
(b) $ c_{1} e^{2t} \left( \matrix 1 \\ 0 \\ 0 \endmatrix \right)
	+ c_{2} e^{t} \left( \matrix 0 \\ 1 \\ 0 \endmatrix \right)
	+ c_{3} e^{t} \left( \matrix 0 \\ 0 \\ 1 \endmatrix \right)$
\medskip 
(c) $ c_{1} e^{2t} \left( \matrix 1 \\ 0 \\ 0 \endmatrix \right)
	+ c_{2} e^{t} \left( \matrix 0 \\ 1 \\ 0 \endmatrix \right)
	+ c_{3} te^{t} \left( \matrix 0 \\ 1 \\ 0 \endmatrix \right) $
\medskip 
(d) $ c_{1} e^{2t} \left( \matrix 1 \\ 0 \\ 0 \endmatrix \right)
	+ c_{2} e^{t} \left( \matrix 0 \\ 1 \\ 0 \endmatrix \right)
	+ c_{3} (te^{t} \left( \matrix 0 \\ 1 \\ 0 \endmatrix \right)
	+ e^{t} \left( \matrix 0 \\ 1 \\ 1 \endmatrix \right) )$
\medskip 
(e) $ c_{1} e^{2t} \left( \matrix 1 \\ 0 \\ 0 \endmatrix \right)
	+ c_{2} e^{t} \left( \matrix 0 \\ 1 \\ 0 \endmatrix \right)
	+ c_{3} e^{t} \left( \matrix 0 \\ 1 \\ 1 \endmatrix \right) $


 

\newpage 
 
\bigskip 
\noindent 
9. Let us consider the first order system
$$ \frac{d {\bold x}}{d t} = A {\bold x} =   \left( \matrix
				1 & -1 \\
				1 & 1 \endmatrix \right){\bold x}.
$$
 Suppose that the matrix $A$ has a complex eigenvalue
$r = 1+ i$ and $\left(\matrix   1\\
- i\endmatrix \right)$ is the corresponding eigenvector. Determine which of
the following is the general solution.
\medskip
(a) $ c_{1} e^{t} (\cos{t} \left( \matrix 1 \\ 0 \endmatrix \right)
	- \sin{t} \left( \matrix 0 \\ -1 \endmatrix \right))
	+c_{2} e^{t} (\sin{t} \left( \matrix 1 \\ 0 \endmatrix \right)
	+ \cos{t} \left( \matrix 0 \\ -1 \endmatrix \right))$

\medskip 
(b) $ c_{1} e^{t} (\cos{t} \left( \matrix 1 \\ 0 \endmatrix \right)
	+ \sin{t} \left( \matrix 0 \\ -1 \endmatrix \right))
	+c_{2} e^{t} (\sin{t} \left( \matrix 1 \\ 0 \endmatrix \right)
	- \cos{t} \left( \matrix 0 \\ -1 \endmatrix \right))$
\medskip 
(c) $ c_{1} e^{t} (\sin{t} \left( \matrix 1 \\ 0 \endmatrix \right)
	- \cos{t} \left( \matrix 0 \\ -1 \endmatrix \right))
	+c_{2} e^{t} (\sin{t} \left( \matrix 1 \\ 0 \endmatrix \right)
	+ \cos{t} \left( \matrix 0 \\ -1 \endmatrix \right))$
\medskip 
(d) $ c_{1} e^{t} \cos{t} \left( \matrix 1 \\ 0 \endmatrix \right)
	+c_{2} e^{t} \sin{t} \left( \matrix 0 \\ -1 \endmatrix \right)$
\medskip 
(e) $ c_{1} e^{t} \left( \matrix 1 \\ 0 \endmatrix \right)
	+c_{2} e^{-t} \left( \matrix 0 \\ -1 \endmatrix \right)$



 

\bigskip 
\noindent 
10. Find the fundamental matrix $\Phi(t)$ satisfying $\Phi(0) = I$
for the linear system
$$ \frac{d {\bold x}}{d t} = \left( \matrix
				0 & 2 \\
				2 & 0 \endmatrix \right){\bold x}.
$$

\medskip 
(a) $ \frac{1}{2}\left( \matrix e^{2t} + e^{-2t}, & 
		 e^{2t} - e^{-2t}  \\
		e^{2t} - e^{-2t}, & 
		e^{2t} + e^{-2t}  \endmatrix \right)
	$
\medskip 
(b)  $ \left( \matrix e^{2t}, &  e^{-2t}  \\
		e^{2t}, & 
		-  e^{-2t}  \endmatrix \right)
	$
\medskip 
(c)  $ \left(  \matrix e^{-2t}, & 
		 e^{2t}  \\
		 e^{-2t}, & 
		- e^{2t}  \endmatrix \right)
	$
\medskip 
(d)  $ \frac{1}{2}\left( \matrix e^{2t} + e^{-2t}, & 
		- e^{2t} + e^{-2t}  \\
		e^{2t} - e^{-2t}, & 
		e^{2t} + e^{-2t}  \endmatrix \right)
	$
\medskip 
(e)  $ \frac{1}{2}\left( \matrix e^{2t} + e^{-2t}, & 
		 e^{2t} - e^{-2t}  \\
		- e^{2t} + e^{-2t}, & 
		e^{2t}  + e^{-2t}  \endmatrix \right)
	$


\newpage 

\bigskip 
\noindent 
11. Given that a fundamental matrix for the linear system
$$ \frac{d {\bold x}}{d t} = \left( \matrix
				-2 & 1 \\
				1 & -2 \endmatrix \right){\bold x}
$$ is 
$$ \Psi(t) = \left( \matrix e^{-3t} &  e^{-t}  \\
		- e^{-3t} & e^{-t}  \endmatrix \right),
$$
find the general solution of 
$$ \frac{d {\bold x}}{d t} = \left( \matrix
				-2 & 1 \\
				1 & -2 \endmatrix \right){\bold x}
	+ \left( \matrix \tan{t} \\ e^{-t} \endmatrix \right)
$$ is

\medskip
\noindent 
(a) $ c_{1} e^{-3t} \left( \matrix 1 \\ -1 \endmatrix \right)
	+ c_{2} e^{-t} \left( \matrix 1 \\ 1 \endmatrix \right)	+ \frac{1}{2}\left( \matrix e^{-3t} &  e^{-t}  \\
		- e^{-3t} & e^{-t}  \endmatrix \right)
	\big\int \left( \matrix e^{3t} & - e^{3t}  \\
		 e^{t} & e^{t}  \endmatrix \right)
		\left( \matrix \tan{t}  \\
		e^{-t}  \endmatrix \right) \, dt.
	$
\medskip
\noindent 
(b) $  c_{1} e^{-3t} \left( \matrix 1 \\ -1 \endmatrix \right)
	+ c_{2} e^{-t} \left( \matrix 1 \\ 1 \endmatrix \right)
	+ \int 	\left( \matrix \tan{x}  \\
		e^{-t}  \endmatrix \right) \, dt.
	$
\medskip
\noindent 
(c)  $  c_{1} e^{-3t} \left( \matrix 1 \\ -1 \endmatrix \right)
	+ c_{2} e^{-t} \left( \matrix 1 \\ 1 \endmatrix \right)
	+ \frac{1}{2}\left( \matrix e^{-3t} &  e^{-t}  \\
		- e^{-3t} & e^{-t}  \endmatrix \right)
	\int \left( \matrix e^{t} & - e^{3t}  \\
		 e^{t} & e^{3t}  \endmatrix \right)
		\left( \matrix \tan{t}  \\
		e^{-t}  \endmatrix \right) \, dt.
	$
\medskip
\noindent 
(d)  $ c_{1} e^{-3t} \left( \matrix 1 \\ -1 \endmatrix \right)
	+ c_{2} e^{-t} \left( \matrix 1 \\ 1 \endmatrix \right) 
	+ \frac{1}{2}\left( \matrix e^{3t} & - e^{3t}  \\
		- e^{-3t} & e^{-t}  \endmatrix \right)
	\int \left( \matrix e^{-3t} &  e^{-t}  \\
		- e^{-3t} & e^{-t}  \endmatrix \right)
		\left( \matrix \tan{t}  \\
			e^{-t}  \endmatrix \right) \, dt.
	$
\medskip
\noindent 
(e)  $ c_{1} e^{-3t} \left( \matrix 1 \\ -1 \endmatrix \right)
	+ c_{2} e^{-t} \left( \matrix 1 \\ 1 \endmatrix \right) 	+ \frac{1}{2}\left( \matrix e^{3t} & - e^{3t}  \\
		- e^{-3t} & e^{-t}  \endmatrix \right)
	\int \left( \matrix e^{-3t} &  e^{-t}  \\
		- e^{-3t} & e^{-t}  \endmatrix \right)
		\left( \matrix \tan{t}  \\
		e^{-t}  \endmatrix \right) \, dt.
	$


\bigskip

\newpage

\noindent 
12.  Consider the almost linear system
$$ \matrix \frac{d x}{d t} =  x-2y +3 y^{2} \\
   \hskip 7pt \frac{d y}{d t} = y -  2xy +3x^{2}. \endmatrix $$

The critical point $(0, 0)$ is
\medskip 
(a) unstable  and
its type for the non-linear system cannot be determined from the 
corresponding linear system.

\medskip 
(b)  an unstable improper node
for both non-linear system and the
corresponding linear system.

\medskip 
(c) unstable for the corresponding linear system and  stable
for the non-linear system.

\medskip 
(d) The stability for the non-linear system cannot be determined
from the corresponding linear system.

\medskip 
(e) an unstable improper node for the corresponding linear system
but an unstable node for the non-linear system.

\bigskip 
\bigskip 
\bigskip 
\bigskip 
\bigskip 

\noindent 
13. For the almost linear system
$$ \matrix \frac{d x}{d t} =  2y -xy - y^{2} \\
   \hskip 15pt \frac{d y}{d t} = -x+3x^{2} -2xy, \endmatrix $$

\medskip 
(a) $(1, 1)$ is an asymptotically stable spiral point.

\medskip 
(b) $(1, 1)$ is either a spiral point or a center. Its stability
can not be determined from the corresponding linear system.


\medskip 
(c) $(1, 1)$ is an unstable saddle point.

\medskip 
(d) $(1, 1)$ is an asymptotically stable node.
 
\medskip 
(e) The type of the critical point $(1, 1)$ cannot be determined
from the corresponding linear system.
 
 
\bigskip 
 

 
\newpage

\noindent 
14. The system
$$ \matrix \frac{d x}{d t} =  -x^{3} + xy^{2}  \\
   \hskip 2pt \frac{d y}{d t} = -3 xy^{2} \endmatrix $$
has $(0, 0)$ as a critical point.
Using $V(x, y) = 3x^{2} + y^{2}$ as a Liapunov function
to test the stability of $(0, 0)$,
what conclusion can we get?

\medskip
(a) $(0, 0)$ is a stable critical point.

\medskip
(b) $(0, 0)$ is a saddle point.

\medskip
(c) $(0, 0)$ is an unstable critical point.

\medskip
(d) No conclusion can be drawn by using this Liapunov function.
We should use another Liapunov function to test the stability
of $(0, 0)$.

\medskip
(e) The stability of $(0, 0)$ cannot be determined by Liapunov's method.




\bigskip
\noindent
15. Suppose $y' = (y+t)^2$, $y(2) = 1$.  Approximate $y(2.2)$ using Euler's
Method, with $h = 0.1$.

(a)  3.5    

(b)  3.2    

(c)  3.3    

(d)  3.4    

(e)  3.6  
  
\bigskip
\noindent
16. If $u_{xx}(x,t) = 2u_{xt}(x,t)$, where $u(x,t) = X(x)T(t)$, which
pair of equations is satisfied separately by $X(x)$ and $T(t)$?

(a)  $X'' + \lambda X' = 0$, $2T' + \lambda T = 0$

(b)  $X'' + \lambda X = 0$, $T'' + 2\lambda T = 0$

(c)  $X' + \lambda X = 0$, $T' + \lambda T = 0$

(d)  $X'' + \lambda X' + X = 0$, $T = 2\lambda$

(e)  there is no solution of form $u(x,t) = X(x)T(t)$  
  
  
\bigskip
\noindent
17.  Suppose that $$ f(x)=
\left\{
\aligned  &0, \quad \quad -2 \le x < 0  \\
                        & 1, \quad \quad 0 \le x < 2. \endaligned
\right.
$$
and $f(x+4) = f(x)$.  
If $a_{0}+\sum_{n=1}^{\infty}(a_{n}\cos(\frac{n\pi}{2}x)+
b_{n}\sin(\frac{n\pi}{2}x))$ is the Fourier series for $f(x)$,  
what is $b_{3}$?

(a)  $\frac{2}{3\pi}$  

  (b)  $\frac{3\pi}{2}$    

(c)  $\frac{1}{6\pi}$

 (d)
$0$    

(e)  $\frac{3\pi}{4}$  
  

\newpage 

\bigskip
\noindent
18. Which is the value does  the Fourier series from Problem 4
at $x = 2$?

(a)  $\frac{1}{2}$   

 (b)  $1$    

(c)  $0$    

(d)  $\frac{1}{4}$ 


(e)  the series does not converge at $x = 2$  
  
\bigskip
\noindent
19.  If $f(x) = x$, $0\leq x < 2$, and $f(x+4) = f(x)$,
what is the value of the even extension of $f(x)$ at $x = 3$?

(a)  $1$    

(b)  $0$    

(c)  $2$    

(d)  $3$    

(e)  $-1$  


\bigskip
\noindent
20.  A bar of metal is located on the $x$-axis between $x = 0$ and $x = 2$.
The bar is insulated, and both ends are maintained at temperature $0$.  At
time $0$, the temperature at position $x$ is given by $f(x) =
\sin(\frac{\pi}{2} x) + 2\sin(\pi x)$, for $0 < x < 2$.  Assume $\alpha = 2$,
so that $u_{xx}(x,t) = 4u_{t}(x,t)$.  Find the
temperature
$u(x,t)$ for $0 < x < 2$ and $t > 0$.

(a)  $\sin(\frac{\pi}{2}x)e^{-\pi^{2}t}+2\sin(\pi x)e^{-4\pi^{2}t}$

(b)  $\sin(\frac{\pi}{2}x)\cos(\pi t) + 2\sin(\pi x)\cos(2\pi t)$

(c)  $\sin(\frac{2}{\pi}x)\cos(\frac{1}{\pi}t) +
2\sin(\frac{1}{\pi}x)\cos(\frac{1}{2\pi}t)$

(d)  $\sin(\frac{2}{\pi}x)e^{-\pi t} + 2\sin(\frac{1}{\pi} x)e^{-2\pi t}$

(e)  $u(x,t) = 0$



  
\bigskip
\noindent
21.  Suppose $f(x)$ is piecewise continuous on the interval $[0,3]$, with
Fourier sine series
$\sum_{n=1}^{\infty}\frac{1}{n^{2}}\sin(\frac{n\pi}{3}x)$.  Find the solution for the wave equation
$u_{xx}(x,t) = u_{tt}$, boundary conditions $u(0,t) = 0$, $u(3,0) = 0$,
for $t\geq 0$, and initial conditions $u_{t}(x,0) = 0$ and $u(x,0) =
f(x)$.

(a)  $u(x,t) =
\sum_{n=1}^{\infty}\frac{1}{n^{2}}\sin(\frac{n\pi}{3}x)\cos(\frac{n\pi}{3}t)$

(b)  $u(x,t) = \frac{2\pi}{3}\sum_{n=1}^{\infty}\frac{1}{n^{2}}
(\sin(n\pi x)\cos(\frac{n\pi}{3}t)$

(c)  $u(x,t) =
\frac{1}{2}\sum_{n=1}^{\infty}\frac{1}{n^{2}}\sin(\frac{n^{2}\pi^{2}}{9}
x)\cos(\frac{n^{2}\pi^{2}}{9}t)$

(d)  $u(x,t) =
\frac{9}{\pi^{2}}\sum_{n=1}^{\infty}\frac{1}{n^{2}}\sin(\frac{n\pi}{3}x)
e^{-\frac{n^{2}\pi^{2}}{9}t}$

(e)  $u(x,t) = 0$  
  




\end