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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 1, Version A} 
\centerline{\bf {September 14, 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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\noindent 
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. A table of Laplace transforms is supplied at the end of the booklet. 
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6. 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} 
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\noindent 
1.  Find the general solution of 
 
$$y''' - y'' - y' + y=0.$$ 
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a.  $C_1e^t+C_2e^{-t}+C_3te^{2t}$. 
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b.  $C_1e^t+C_2\sin t + C_3 \cos t $. 
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c. $C_1te^t+C_2e^{t}+C_3 e^{-t}$. 
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d.  $C_1te^t +C_2 \sin t +C_3 \cos t$. 
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e. $C_1e^t+C_2t e^{-t} +C_3 e^{-t}$. 
 
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\noindent 
2. Which of the following is a solution of the equation 
$$y'''+8y''+25y'=0.$$ 
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a. $e^{3t} \cos  5t.$ 
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b. $e^{4t}\sin  3t.$ 
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c. $e^{-4t}\cos 3t.$ 
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d. $ e^{-3t}\cos 4t .$ 
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e. $ e^{-5t}\cos 4t .$ 
 
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\noindent 
3. Find the form of a particular solution of 
$$y'''-3y''+ 3y' - y=e^{t}.$$ 
(Hint, 1 is a triple root of the characteristic polynomial $Z(r) = (r-1)^3$.)
 
a.  $ Ce^{t}$. 
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b.  $C t e^{t}.$ 
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c.  $e^{-t}(A_0 t^3 + A_1 t^2 + A_2 t + A_3) $. 
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d.  $A_0 t^3 e^t$. 
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e. $ C(A_0 t + A_1)$. 
 
 
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\noindent 
4. Find the  Wronskian of the functions $\{e^x, e^{2x},e^{3x } \}$. 
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a. $  e^{6 x}$ 
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b. $  3e^{ 6x}$ 
 
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c. $ 6 e^{ 6x}$. 
 
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d. 0 
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e. $2e^{6x}$ 
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5.      Let $f(t)$ be the function defined by  
$$ f(t) = \cases   
   & t-2, \quad \text{if} \  
                    t \le 2 \\ 
   & 0,  \qquad\quad\  \text{if} \ t > 2,   
  \endcases 
$$ 
%% $$f (t)  = \left\{ \aligned 
%% &1,\quad 
%%  0< t<2 \\ 
%% &0,\quad t\ge 2. \endaligned  $$ 
 
 Find the  Laplace 
transform   $F(s)$  of $f$. 
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a.	$\dsize \frac{1}{s^2}  - \frac{1}{s} - \frac{e^{-2s}}{s}$. 
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b.	$\dsize \frac{1}{s^2}  - \frac{2}{s} - \frac{e^{-2s}}{s^2}$. 
 
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c.   $ \dsize \frac{2}{s^2}  - \frac{1}{s} - \frac{e^{-s}}{s}$. 
 
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d.	$\dsize \frac{1}{s}   - \frac{e^{-2s}}{s^2}$. 
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e. $\dsize \frac{1}{s^2}  (1 - e^{-2s})$. 
 
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\noindent 
6. Find the  inverse Laplace transform of 	$\dsize F (s) =  \frac { 4}{s^2 - 
6s + 10}$        
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a.  $4 e ^{3 t }\sin t.$  \qquad
  b.  $3 e ^{4 t }\sin t.$ 
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c.  $4 e ^{3 t }\cos 3t.$ 
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d.  $4 e ^{3 t }\cos t.$ 
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e.  $4 e ^{3 t } t.$ 
 
 
 
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7.	 Find the Laplace transform of the function 
$ f(t) = \delta(t-1)(t^{99} + 99)$. 
 
%% $$ f(t)=  \left\{\aligned  &0, \quad \quad t<1 \\ & t^2-2t,\quad t\ge 
%% 1.\endaligned $$ 
 
 
a.	 $99 e^{-s}.$ 
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	b. $100 e^{- 99 s}.$ 
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	c.	 $99 e^{s}.$ 
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	d.	$s^{-99} e^{-s}.$ 
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	e.	 $100 e^{-s}.$ 
 
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\noindent 
8.   Find the inverse Laplace transform of  $F(s)=\dsize \frac 
{se^{-2s}}{s^2-9}$. 
 (Hint: use the a table of Laplace transforms supplied at at the end of the booklet.)
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a.  $u_2(t)\cosh 3 (t-2) $. 
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b.  $  u_2(t)\sinh 3(t-2)$. 
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c.  $u_3(t)\sin 2 (t-2) $. 
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d.   $u_9(t)\sinh 2 (t-1) $. 
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 e. $u_3(t)\cos 2(t-3)$. 
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\noindent 
9.    Find the 
 solution to the following initial value problem 
$$y'''+ 8y''+ 8y' + 9y =1;\quad y(0)=1,\ y'(0)=  y''(0) = 0. $$ 
 
a.  $y = \frac 19 $. 
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 b. $y=\Cal L^{-1}(\dsize\frac { s^{-1} + s^2 + 8s + 8 }{s^3 + 8s^2+8s+9})$. 
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c.  $y=\Cal L^{-1}(\dsize\frac { s^{-1} + s^2  }{s^3 + 8s^2+8s+9})$. 
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  d. $y=\Cal L^{-1}(\dsize\frac { 1 + s^2 + 8s }{s^3 + 8s^2+8s+9})$. 
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e.  $y=\Cal L^{-1}(\dsize\frac { 1- s^2 - 8s + 8 s^3 }{s^3 + 8s^2+8s+9})$. 
 
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\noindent 
{\bf PART B} 
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10. Solve the initial value problem 
$$y''+ 9 y= 9 u_\pi(t); \quad  y(0)=0, \ y'(0)=0.$$ 
 
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11. a.  Find  a  particular solution of the equation 
$$y'''- 5y'' + 4 y' = 10 e^{-t} + 16t.$$ 
 (Note that $\{ 1, e^{4t}, e^{t} \} $ form a fundamental set of solutions
for the complementary equation.)

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b. Find the general solution for the equation. 
 
 
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12.  a.  Find  a fundamental  set of solutions of the equation 
$$y''' - y'= 0.$$ 
 
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b. Using the method of variation of parameters, find a particular solution of  
the equation $$y''' - y'= g(t).$$ 
(Hints:  Since you do not know g(t), you must indicate what to do in terms of it.) 
 
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