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\begin{document}




\begin{minipage}[c]{6in}
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 7 PAGES with 11 PROBLEMS.  The point value for each 
multiple choice problem is 8 points. 
Problems 9, 10 and 11 are worth12 points each giving a total of 100 points.
Use the back of the test pages  for scratch work.\\
3.   On all partial credit problems, show your work, indicating clearly 
 how you arrived at your answer. 
 The points you receive for problems 9, 10 and 11 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. \\
4. At the end of the test  the table of Laplace transforms is attached.
\end{minipage}}
\answersheetfootline{\large\bf 
\fbox{\rule[-2mm]{0mm}{8mm}Sign your name:\hspace{6cm}}
\hspace{6cm}}
\twocolumn{12} 
\date{{\large Sept. 25, 1992}}
\lastversion=3 \noprofessor \nosection \vertmultchoice=2.5in
\multiplechoiceskip=0.8cm \let\ansborder=\boxborder \boxwidth=1cm
\boxdepth=0.4cm

\pagenumberstyle=1
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\problem  %%{\bf Fall 92, Test 1}

Find the general solution of the differential equation
$$y'' + 10y' + 26y = 0$$
\correct $y = e^{-5x}(c_1\cos(x) \! + \! c_2\sin(x))$
\wrong $y = c_1\cos({3\over2}x) \! + \! c_2\sin({3\over2}x)$ 
\wrong $y = c_1e^{-5x} \! + \! c_2e^{5x}$ 
\wrong $y = e^{10x}(c_1\cos(x) \! + \! c_2 \sin(x))$
\wrong $y = c_1e^{-4x}\cos(x) \! + \! c_2xe^{-4x}\sin (x)$  

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\problem   %%{\bf Fall 92, Test 1}
 
Assume ${\cal L}\{ f(t)\} =\cos (s) $. Then
${\cal L}^{-1}\{\cos (s+2)\} = $


\correct $e^{-2t}f(t)$
\wrong  $e^{2t}f(t)$
\wrong  $f(t-2)$
\wrong  $e^{3t}f(t-3)$
\wrong  $f(t+3)$


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\problem   %%{\bf Fall 92, Test 1}

 Denote with $u_{c}(t)$ the unit step function.
 Find the Laplace transform of $f(t)=u_{3}(t)(t-5).$

\correct $F(s)=\frac{e^{-3s}-2se^{-3s}}{s^{2}}$
\wrong  $F(s)=\frac{1+e^{-3s}}{s}$
\wrong  $F(s)=\frac{1}{s^{3}}$
\wrong  $F(s)=\frac{e^{-3s}}{s^{2}+1}$
\wrong  $F(s)={s^{-2}}+e^{-3s}$

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\problem   %%{\bf Fall 92, Test 1}
 
Calculate the Wronskian  determinant  of the functions: 
$$\{\cosh (x), e^{x} , e^{-x}\}. $$


\correct $0$
\wrong  $e^{3x}$
\wrong  $\cosh (x) \sinh (x)$
\wrong  $1$
\wrong  $\cosh (x)$

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\problem   %%{\bf Spring 92, Test 1 and Fall 92, Test 1}

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

\correct $x \cos (x)$
\wrong  $xe^{-2x}$
\wrong  $x \cos (2x)$
\wrong  $x^{2} \sin (x)$
\wrong  $x^{2}e^{x}$
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\problem   %%{\bf Fall 91, Quiz1 also Fall 92, Test1}

Let $y(x) = u_1(x)\cos(x) + u_2(x)\sin(x)$ be a particular solution of
$$y'' + y = \tan(x),\qquad 0 < x < {\pi\over2}$$
obtained by the method of variation of parameters. Find $u_2$.

\correct $u_2 = -\cos(x)$ 
\wrong $u_2 = \sin(x)$ 
\wrong $u_2 = \sec^2(x)$ 
\wrong $u_2 = -\sec(x)$ 
\wrong $u_2 = \cot(x)$ 


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\problem   %%{\bf Spring92 and Fall 92, Test 1}
 Find the inverse Laplace transform of the function
$$ F(s) = \frac{2 e^{-4s}}{s^{2} + 4}\;\;.$$

\correct $u_{4}(t)\sin  2(t-4)$
\wrong  $u_{4}(t) e^{-2(t-4)}$
\wrong  $u_{4}(t)\cos  2(t-4)$
\wrong  $u_{4}(t) (t-4)$
\wrong  $u_{4}(t) e^{-4(t-2)}$


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\problem   %%{\bf Fall 92, Test 1}
 Solve the initial value problem
$$ y'' + 2y' + y = 0;
 \hspace{1cm} y(0)=-3,\; y'(0)=4.$$

\correct  $y = e^{-x}(x-3)$
\wrong $y = (-3)e^{x} +xe^{x}$
\wrong  $y =(-3)e^{x} +xe^{-x} $
\wrong  $y = (7x-3)e^{-x}$
\wrong  $y = (7x-3)e^{x}$

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\problem   %%{\bf Fall 92, Test 1}
 (12 pts.) 
Use the Laplace transform to solve the initial value problem: 
$$y'' + 4y' + 4y = t e^{-2t},\;\;\;\;y(0) = 0,\;\;y'(0) = 1.$$

\vspace*{15cm}

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\problem   %%{\bf Fall 92, Test 1}
 (12 pts.) 
Use the method of undetermined coefficients to solve the initial 
value problem:
 $$y'' + 10y' + 25y= 108xe^{x} + 25 \hspace{1cm} y(0)=0,\; y'(0)=0.$$ 

\vspace*{15cm}

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\problem   %%{\bf Fall 92, Test 1}
 (12 pts.)  Solve the initial value problem
$$ y'' + 4y = g(t);
 \hspace{7mm} y(0)=0,\; y'(0)=0,\hspace{4mm}
g(t):= \left\{ \begin{array}{ll}
0 & 0\leq t< 1  \\ 1 & 1\leq t <2 \\ 0 &2\leq t<\infty \end{array} \right. $$

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