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\begin{document}








\noindent
1. This examination contains twelve problems worth a total of  100  points.
The test will be exactly
60 minutes in length.


(Part A: 6 x 10  points = 60 points,
 Part B: 2 x 15 points = 30 points and you begin with 10 points)
\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.}
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\noindent
3. On problem   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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\noindent
4.  Calculators, books and notes are
not allowed.
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\noindent
5. A table of Laplace transforms is supplied at the end of the booklet.
\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}":
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\hskip .7 in\vrule height .10pt depth 0 pt width 2 in
\bigskip
\bigskip
\bigskip
\centerline{\bf GOOD LUCK}



\newpage
\noindent
{\bf PART A}
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\noindent
1.  Find the general solution of

$$y'''-4y''+5y'-2y=0.$$
\medskip
a.  $C_1e^t+C_2e^{2t} $.
\medskip
b.  $C_1e^t+C_2te^{ t} +C_3e^{2t}$.
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c. $C_1e^t+C_2e^{-t}+C_3e^{2t}$.
\medskip
d.  $C_1e^t +C_2te^t+C_3 e^{-2t}$.
\medskip
e. $C_1 \cos t+C_2\sin t+C_3e^{-2t}$.

\bigskip
\noindent
2. Consider the initial value problem
$$  t y'' +2y'+(\cos t )y =t^2. $$  Where  are the intervals in which
solutions are sure to exist?
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a. $-\infty<t<\infty$.
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b. $t>0$ only.
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c. $t<0$ only.
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d. $t>0$ or $t<0$.
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e. $t=0$.

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\noindent
3. Consider the nonhomogeneous equation
$$y^{\text{iv}} +2y''+y =3\cos t -14t^2$$  Find the suitable form of a
particular solution if the
method of undetermined coefficients is to be used.

\medskip
a. $t^2(A\cos t+B\sin t)+Ct^2+Dt+E$.
\medskip
b.  $t(A\cos t+B\sin t)+Ct^2+Dt+E$.
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c. $ A\cos t+B\sin  t +Ct^2 $.
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d. $ A\cos  t +Bt^2+Ct+D$.
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e. $t^2(A\cos 2t+B\sin 2t)+t(Ct^2+Dt+E)$.


\newpage
\bigskip
\noindent
4. Find the general solution  of the equation
$$y''-y=6e^{2t}.$$
\medskip
a. $y(t)=C_1e^t+C_2e^{-t}+e^{2t}$.
\medskip
b. $y(t)=C_1e^t+C_2e^{-t}+2e^{2t}$.

\medskip
c. $y(t)=C_1e^t+C_2e^{-t}+te^{2t}$.
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d. $y(t)=C_1e^t+C_2e^{-t}-e^{2t}$.
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e. $y(t)= 2e^{2t}$.

\bigskip
\noindent
 5.    Which set of functions is linearly dependent?
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a.	 $f_1(t)=1,f_2(t)=t$.
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b.	$f_1(t)=t+1,f_2(t)=t$.

\medskip
c.  $f_1(t)=t+1,f_2(t)=t,f_3(t)=t^2 $.

\medskip
d.	$f_1(t)=t+1,f_2(t)=t,f_3(t)=2t+3 $.
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e. $f_1(t)=t+1,f_2(t)=t,f_3(t)=t^2+t $.

\bigskip
\noindent
6. Find the   Laplace transform of 	$$f(t)=\delta(t-\pi)t^2.$$
\medskip
a.  $\frac {2}{s^3} e^{- \pi s}$.
\medskip

b.   $-\pi^2e^{\pi s}$ .
\medskip
c.  $\pi^2$.
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d.  $\pi^2e^{\pi s}$.

\medskip
e. $\pi^2e^{ -\pi s}$.



\newpage


\bigskip

\noindent
7.	 Find the Laplace transform of the function

 $$f(t)=  \cases &0, \qquad \quad 0< t<1 \\ & t^2-2t,\quad t\ge 1.\endcases$$


a.	 $\dsize e^{-s}(\frac 2{s^3}-\frac 1s).$
\medskip
	b.	 $\dsize e^{-s}(\frac 2{s^3}+2\frac 1s).$
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	c.	 $\dsize e^{- s}(\frac 2{s^3}-1).$
\medskip
	d.	 $\dsize e^{-s}(\frac 1{s^2}-\frac 1s).$
\medskip
	e.	 $\dsize  \frac {2e^{-s}}{s^3}+1 .$

\bigskip
\noindent
8.   Find the inverse Laplace transform of  $f(s)=\dsize \frac {s
}{s^2-2s+5}$.

\bigskip
a. $e^t \cos 2t $
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b.   $e^t \cos 2t+e^t\sin 2t$
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c.  $e^t \cos 2t-e^t\sin 2t$
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d.  $e^{-t }\cos 2t+e^{-t}\sin 2t$
\medskip
 e.   $e^t \cos 2t+\frac 12 e^t\sin 2t$

\newpage
\bigskip
\noindent
 9.    Find the
 solution to the following initial value problem
$$y^{\text{iv}}+y''+y=1;\quad y(0)=0,\ y'(0)=0,y''(0)=0, y'''(0)=0  $$

a.  $y=1 $.
\medskip
 b. $y=0$.
\medskip
c.  $y=\Cal L^{-1} (\dsize\frac { s }{s^4+s^2+1} )$.
\medskip
  d. $ y= \Cal L^{ -1}(\dsize\frac { 1}{s^4+s^2+1}) $.
\medskip

e.  $y=\Cal L^{ -1}(\dsize\frac { 1}{s(s^4+s^2+1)})$.






\bigskip

\noindent
10.    Find the
 solution to the following initial value problem
$$ y''+y=\delta(t-\pi);\quad y(0)=0,\ y'(0)=0.  $$

a.  $y=u_\pi (t)\sin (t-\pi) $.
\medskip
 b. $ y= e^{-\pi t}\sin t$.
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c.  $y=u_\pi(t) \cos (t-\pi)$.
\medskip
  d. $ y= \sin t $.
\medskip

e.  $y= 0$.

\newpage
{\bf PART B}
\newline
\noindent
 11.  Use the method of variation of parameters to  find the general
solution of the equation
$$y'''-y'=2e^t.$$

\newpage
\newpage

12.  Solve the initial value problem
$$y'' +4y=4u_\pi(t); \quad  y(0)=0, \ y'(0)=1.$$

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