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\begin{document}







\title{{\large\bf Math 325, Final}}
\comment{ \large 
\begin{minipage}[c]{5.8in}
1. Please cross \fbox{$\times $} the correct answers. \\
2.  This test will be exactly 120 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 9 PAGES with 15 PROBLEMS.  The
point value for each problem is 10 points with a total of 150 points.  
The blank sheet and the backs
of the test pages are for scratch work.\\
3. {\bf  On all partial credit problems, show your work, indicating clearly 
 how you arrived at your answer. 
 The points you receive for problems 12 to 15 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.}
\end{minipage}} 
\answersheetfootline{\large\bf 
\fbox{\rule[-2mm]{0mm}{8mm}Sign your name:\hspace{6cm}}
\hspace{6cm}}
\twocolumn{8} 
\date{{\large Dec. 19, 1991}}
 \noprofessor \nosection \vertmultchoice=2.5in
\multiplechoiceskip=0.8cm \let\ansborder=\boxborder \boxwidth=1cm
\boxdepth=0.4cm

\pagenumberstyle=1
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\problem Denote by $\delta (t)$ the Dirac delta function.
 Calculate the following improper integral:
 $$\int_{0}^{\infty}t e^{t^{2}-9} \delta (t-3)dt$$

\correct $3$
\wrong  $\delta (3)$
\wrong  $e^{\smallfrac{2}{3}}$
\wrong  $0$
\wrong  $e$


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\problem Solve the initial value problem
$$ y''+y=\delta (t-\pi)-\delta (t-2\pi); \hspace{1cm} y(0)=0,\; y'(0)=0.$$

\correct $u_{\pi}(t)\sin (t-\pi) - u_{2\pi}(t)\sin (t-2\pi)$
\wrong  $\frac{1}{3}u_{\pi}(t)\sin (t-\pi) - \frac{2}{3} u_{2\pi}(t)\sin (t-2\pi)$
\wrong  $u_{2}(t)(t-5)+u_{\pi}(t)\cos (t-\pi) - u_{2\pi}(t)\sin (t-2\pi)$
\wrong  $\frac{1}{2}u_{\pi}(t)\cos (t-\pi) - u_{2\pi}(t)\sin (t-2\pi)$
\wrong  $\frac{1}{3}u_{\pi}(t)\sin (t-\pi) - u_{2\pi}(t)\sin (t-2\pi)$
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\problem
Let $y(x) = u_1(x)\cos(x) + u_2(x)\sin(x)$ be the solution of $$y'' + y = \sin(x)\cos(x)$$
given by the method of variation of parameters. Find $u_1$. 
\correct  $u_1 = -{1\over3}\sin^3(x)$
\wrong {$u_1 = -\cot(x)$}
\wrong $u_1 = -\sin^2(x)\tan(x)$ 
\wrong {$u_1 = \tan(x)$}
\wrong {$u_1 = {1\over2}\cos^2(x)$}


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\problem If ${\cal L}\{ f(t)\} =\frac{1}{\sqrt{s^{2}+9}} $ and $f(0)=7$.
 Then what is ${\cal L}\{ f'(t)\}  $


\correct $\frac{1}{\sqrt{s^{2}+9}}-7$
\wrong  $\frac{1}{\sqrt{s^{2}+9}}+7$
\wrong  $s\sqrt{s^{2}+9}+7$
\wrong  $\sqrt{\frac{1}{s-7}}$
\wrong  $\sqrt{\frac{7}{s+2}}$
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 \problem 
 Solve the initial value problem $x' = \left( \hspace{-2mm}
\begin{array}{cc} 1 & -4 \\ 4 & -7\\ \end{array} \hspace{-2mm} 
\right)x, \, x(0) =\left( \hspace{-2mm}
\begin{array}{c} 3 \\ 2\\ \end{array} \hspace{-2mm} \right)$.

\correct $x(t) = \left( \hspace{-2mm} \begin{array}{c} 3 + 4t \\ 2 + 4t \\ \end{array} 
\hspace{-2mm} \right)
e^{-3t}$

\wrong $x(t) = \left( \hspace{-2mm} \begin{array}{c} 3 \\ 5 \\ \end{array} 
\hspace{-2mm} \right) + \left( \hspace{-2mm}
\begin{array}{c} t \\ t \\ \end{array} \hspace{-2mm} \right) e^{-3t}$

\wrong $x(t) = \left( \hspace{-2mm} \begin{array}{c} 3 \\ 2 t\\ \end{array}
 \hspace{-2mm} \right) + 
\left( \hspace{-2mm} \begin{array}{c} 1 \\ 1 t\\ \end{array} \hspace{-2mm} \right)e^{-3t}$

\wrong $x(t) =  \left( \hspace{-2mm} \begin{array}{c} 3- t \\ 2 + t\\ \end{array}
\hspace{-2mm} \right) e^{3t}$


\wrong $x(t) = \left( \hspace{-2mm} \begin{array}{c} 1 \\ 4 t\\ \end{array} 
\hspace{-2mm} \right)e^{3t} +
 \left( \hspace{-2mm}
\begin{array}{c} -4 \\ -7 \\ \end{array} \hspace{-2mm} \right)e^{3t}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \problem 
 Consider the system
$x' = \left( \hspace{-2mm} \begin{array}{cc}1 & 1\\ 1 & 1\\ 
\end{array} \hspace{-2mm} \right) x$. The fundamental matrix $\Phi(t)$
satisfying $\Phi(0)=I$ is:


\correct $\frac{1}{2}\left( \hspace{-2mm} 
\begin{array}{cc}e^{2t}+1&e^{2t}-1 \\ e^{2t}-1&e^{2t}+1\\ 
\end{array} \hspace{-2mm} \right)$
\wrong $\left( \hspace{-2mm} \begin{array}{cc}1+t&e^{t}\\ e^{2t}&e^{t}\\ 
\end{array} \hspace{-2mm} \right)$
\wrong $\frac{1}{3}\left( \hspace{-2mm} 
\begin{array}{cc}2e^{2t}+1&e^{2t}-1 \\ e^{2t}-1&e^{2t}+2\\ 
\end{array} \hspace{-2mm} \right)$
\wrong $\frac{1}{2}\left( \hspace{-2mm} 
\begin{array}{cc}e^{2t}+1&2t^{2}-t \\ e^{2t}-1&2t^{2}-t\\ 
\end{array} \hspace{-2mm} \right)$
\wrong $\left( \hspace{-2mm} 
\begin{array}{cc}1&e^{2t}-1 \\ e^{2t}-1&1\\ 
\end{array} \hspace{-2mm} \right)$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
Let $f$ and $g$ be functions defined on $[- \pi, \pi]$ with $f$ odd and $g$
even.  Then
\correct The Fourier Series of $fg$ consists only of sine terms.
\wrong The Fourier Series of $fg$ consists only of cosine terms.
\wrong The Fourier Series of $fg^2$ consists only of cosine terms.
\wrong The Fourier Series of $f^2g$ consists only of sine terms.
\wrong The Fourier Series of $f^2g^2$ consists only of sine terms.

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\problem
Suppose $f(x)$ satisfies the differential equation $y^{\prime } =y-4;\;
y(0)=5$.
Use Euler's method with stepsize $h=1$ to estimate $f(3)$.\hspace{2cm}
(3 iterations).  

\correct $12$
\wrong $0$
\wrong $5$
\wrong $14$
\wrong $4$

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\problem
Consider the function $f(x) = -x; -1 \leq x < 1; f(x+2) = f(x).$
Find the Fourier Series expansion of $f(x).$

\correct $f(x) = \frac{2}{\pi} \sum_{n=1}^{\infty} \frac {(-1)^n}{n} \sin n
\pi x$
\wrong $f(x) = \sum_{n=1}^{\infty} \frac {2}{n \pi} \sin n \pi x.$
\wrong $f(x) = \sum_{n=1}^{\infty} \frac {(-1)^n}{n \pi} \cos n \pi x.$
\wrong $f(x) = \frac {3}{ \pi} \sum_{n=1}^{\infty} \frac {(-1)^{n+1}} {n}
\cos n \pi x.$
\wrong $f(x) = -x \cdot \sin n \pi x.$


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

\correct $0$
\wrong $\pi i x$
\wrong $\frac{2}{3}\pi x$
\wrong $2e^{x}$
\wrong $e^{2ix}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \problem 
 If $\left( \hspace{-2mm} \begin{array}{c}3\\ 2\\ 
\end{array} \hspace{-2mm} \right)$ is an eigenvector of the symmetric matrix
$\left( \hspace{-2mm} \begin{array}{cc}a&b\\ b&c\\ 
\end{array} \hspace{-2mm} \right)$ then which of the 
following vectors is  an eigenvector as well?


\correct $\left( \hspace{-2mm} \begin{array}{c}-2\\ 3\\ 
\end{array} \hspace{-2mm} \right)$
\wrong $\left( \hspace{-2mm} \begin{array}{c}-3\\ 2\\ 
\end{array} \hspace{-2mm} \right)$
\wrong $\left( \hspace{-2mm} \begin{array}{c}4\\ 2\\ 
\end{array} \hspace{-2mm} \right)$
\wrong $\left( \hspace{-2mm} \begin{array}{c}0\\ 0\\ 
\end{array} \hspace{-2mm} \right)$
\wrong $\left( \hspace{-2mm} \begin{array}{c}3\\ 3\\ 
\end{array} \hspace{-2mm} \right)$

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%%%%%%%%%%%%%%%%%%%%%%%%       PARTIAL CREDIT       %%%%%%%%%%%%%%%%%%%%%%%
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\problem
Consider the non-linear system:
$$\begin{array}{clcr}
x' & = & x+y \\
y' & = & \sin (x-y)  \end{array}$$ 

a.  Show that the origin $(0,0)$ is a critical point for this system.

b.  Determine the type and stability for this equilibrium point.

c.  Find all other critical points for this system.

\vspace*{6cm}
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\problem 
Find the inverse Laplace transform of:
$$F(s)=\frac{s-2}{s^{2}-4s+13}$$

\vspace{6cm}



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\problem

Find the condition satisfied by the positive real eigenvalues $\lambda$ 
of the following boundary value problem:
$$y''  +\lambda y=0\hspace{1cm} y(0)=0\hspace{5mm} y(\pi)+y'(\pi)=0.$$


\vspace{6cm}



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\problem 
A metal wire  100cm long has a uniform temperature
of $30^{\circ}C$. At time $t=0$ (today, Thursday Dec. 19 at 15:45pm
E.S.T.) one end is held at a constant temperature
of $10^{\circ}C$ and the other
end is held at a constant temperature of $70^{\circ}C$.
If  no heat is lost along the wire then:

a) What will be the approximate temperature distribution $v(x), \;
0\leq x\leq 100$ on Christmas eve?

b) Using $\alpha^{2}=4$ in the heat equation find the
temperature distribution $u(x,t)$. (Write down $u(x,t)$
as a series and then express the coefficients in this series
as integrals.)

c) (5 points Christmas bonus!) Evaluate these coefficients
and get a series expansion of $u(x,t)$ valid for
$0\leq x \leq 100$, $t>0$.

\vspace*{17cm}

{\huge $$ Merry\;  Christmas!!! $$}

\vspace*{15cm} 



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 \end{document}


