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\begin{document}







 
\title{{\large\bf Math 325, Test 1}}
\comment{ \large 
\begin{minipage}[c]{6.2in}
1. Please cross \fbox{$\times $} the correct answers. \\
2.  This test will be exactly 50 minutes in length.  
When you are told to begin, but not before, glance 
through the entire test and put your name on each
page.  It is YOUR RESPONSIBILITY to make sure your 
test consists of 5 PAGES with 10 PROBLEMS.  The
point value for each problem is 10 points with a total of 100 points.  
The blank sheet and the backs
of the test pages are for scratch work.\\
3.  On all problems, show your work, indicating clearly 
in the space provided how you arrived at your answer. 
 The points you receive for problems 6 to 10 will depend on the
extent to which the work you show convinces the grader 
(at the time of grading) that you did all or
part of the problem correctly,
 whether or not you write down the correct answer.
\end{minipage}} 
\answersheetfootline{\large\bf 
\fbox{\rule[-2mm]{0mm}{8mm}Sign your name:\hspace{6cm}}
\hspace{6cm}}
\twocolumn{12} 
\date{{\large Sept. 25, 1991}}
\lastversion=2 \noprofessor \nosection \vertmultchoice=2.5in
\multiplechoiceskip=0.8cm \let\ansborder=\boxborder \boxwidth=1cm
\boxdepth=0.4cm

\pagenumberstyle=1
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\problem
Find the general solution to
$$y'' - 4y' + 5y = 0$$
\correct
{$y = e^{-3x}\left(c_1\cos(x) + c_2\sin(x)\right)$}
\wrong
{$y = c_1\cos({3\over2}x) + c_2\sin({3\over2}x)$}
\wrong
{$y = c_1e^{-3x} + c_2e^{3x}$}
\wrong
{$y = c_1e^{+2x}\cos(x) + c_2e^{+2x}\sin(x)$}
\wrong
{$y = c_1e^{-2x}\cos(x) + c_2xe^{-2x}\sin(x)$}

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\problem Assume the characteristic equation of a sixth--order
differential equation is $(r^{3}-1)(r-1)^{2}=0$.
Which one of the following functions is a solution? (Only one is.)

\correct $\cosh (x)$
\wrong  $x \cos (\frac{\sqrt{3}}{2} x )$
\wrong  $e^{\smallfrac{x}{2}}\sin x$
\wrong  $xe^{-x}$
\wrong  $x^{2}e^{x}$




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\problem Assume ${\cal L}\{ f(t)\} =\sqrt{\frac{\pi}{s-2}} $. Then
${\cal L}\{e^{-4t}f(t)\} = $


\correct $\sqrt{\frac{\pi}{s-2}}$
\wrong  $e^{-2s}\sqrt{\frac{\pi}{s}}$
\wrong  $e^{-2s}\sqrt{\frac{\pi}{s-2}}$
\wrong  ${\frac{\sqrt{\pi}}{s-2}}$
\wrong  $\sqrt{\frac{\pi}{s+2}}$




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\problem Denote with $u_{c}(t)$ the unit step function.
 Find the Laplace transform of $f(t)=t+u_{1}(t)(t-1).$

\correct $F(s)=\frac{1-e^{-s}}{s^{2}}$
\wrong  $F(s)=\frac{1+e^{-s}}{s^{2}}$
\wrong  $F(s)=\frac{1}{s^{2}}$
\wrong  $F(s)=\frac{2+3s}{s^{2}+1}$
\wrong  $F(s)={s^{-2}}+e^{-2s}$
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\problem Denote with $\delta (t)$ the Dirac delta function.
 Calculate the Laplace Transform of:
 $$f(t) = t Sin \frac{\pi}{2} t \delta (t-1)$$

\correct $\frac{2}{s^2}$
\wrong  $\frac{\delta (2)}{s}$
\wrong  $\frac{1}{\int{s}}$
\wrong  $\frac{e^{-s}}{s}$
\wrong  $e^{-s}$


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\problem Solve the initial value problem
$$ y^{(4)}=\delta (t-5);
 \hspace{1cm} y(0)=0,\; y'(0)=0,\; y''(0)=0,\; y'''(0)=1.$$

\correct $\frac{1}{6}u_{5}(t)(t-5)^{3}$
\wrong  $\frac{1}{6}u_{5}(t)(t-5)^{3}$
\wrong  $u_{2}(t)(t-5)^{3}+(t-5)^{2}$
\wrong  $\frac{1}{2}u_{5}(t)(t-5)^{2}+ \frac{(t)^{3}}{6}$
\wrong  $\frac{1}{3}u_{6}(t)(t-5)^{3}$


\vspace*{5cm}

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\problem Solve the initial value problem:
$$ y^{(4)}-16y=0; \hspace{1cm}
y(0)=1,\; y'(0)=0,\; y''(0)=1,\; y'''(0)=0.$$

\vspace*{4cm}

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\problem Calculate the Wronskian, i.e. the determinant of the Wronskian
matrix, of the functions: $$\{ e^{ix}, \sin(x), \cos(x)\}. $$

\vspace*{7cm}

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\problem Find the inverse Laplace transform of:
$$F(s)=\frac{e^{-s}}{s^{2}+2s-3}$$

\vspace*{-2cm}

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\problem  Consider the differential equation:
$$y''' -2y''  = 4e^{2x}\sin (x).$$
The solution of the homogenous equation is given by:
$$y=c_{1}+c_{2}x + c_{3}e^{2x}.$$
a) Assume $y_{p}(x)=u_{1}(x)+u_{2}(x)x + u_{3}(x)e^{2x}$ is a
particular solution. Use the method of ``Variation of Parameters"
to write down a system of equations satisfied by 
$u_{1}(x),u_{2}(x),u_{3}(x)$.

b)Solve for $u_{3}(x)$.

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