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{\large
\begin{center}
{\bf MATH 220 \hspace{0.4in} Combined Final Exam \hspace{0.4in} 
6pm-8pm 06 May 1999 \hspace{0.4in} Barston \& Hanson}\\
\end{center}
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{\bf Directions:}  Answer {\bf All Questions} and show {\bf All Work} in the
{\bf Exam Booklet} provided.  Write your {\bf Name}, {\bf Social Security
Number}, and {\bf Discussion Section Hour/Day} on the Exam Book Cover Page.
Start each new question at the {\bf top} of a new page and {\bf box} your
final answer.  Each of 8 questions is worth 25 points.
{\bfseries Keep your eyes on your own work and keep your own work covered.}
A {\it Table of Laplace Transforms and a table of Integrals are provided}, but
must be returned at the end of the exam.  You must also return your 
{\bf Formulae Sheet} at the end.
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{\centering\underline{~\hspace*{6.85in}~}}
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\begin{enumerate}
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\item
Obtain the solution \eqline{y(x)} to each of the following initial value 
problems:
\begin{enumerate}
\item \eqline{y'(x) = 2y^2 -2xy^2, ~~ y(0) = \Frac{1}{4}} 
\\*[1ex]
\item \eqline{xy'(x) + 2y(x) = x^2 - 3x + 1, ~~ x > 0, ~~ y(1)=\Frac{1}{4}}
\end{enumerate}
%
\item
Find the general solution of 
\eqlg{x^2y''(x) - 5xy'(x) +  9y(x) = x^3, ~~ x>0}
%
\item Use the \underline{method of undetermined coefficients} to find the 
the general solution of the ODE:
\eqlg{y''(x)+2y'(x) + y(x) = 4 - e^{-x} + 2 \cos(x)} 
%
\item 
Find \underline{\underline{all}} the values of the real number 
\eqline{\lambda > 4} for which the boundary value problem 
\eqlg{\mbox{\bf ODE:} && y''(x) - 4 y'(x) +\lambda y(x) = 0, 
~~ 0 \leq x \leq 1\\
&&\mbox{\bf BC:} ~~ y(0) = 0, ~~ y(1) = 0}
has nontrivial solutions, \eqline{y(x)}, and find these solutions.
%
\item Evaluate:
\begin{enumerate}
\item \eqline{\LC\left[e^{\DSB -4t}\sin(3t) + (t-2)^2e^{\DSB -3t}\right](s)}
\\*[1ex]
\item \eqline{\LCI\left[\Frac{s}{(s+4)^2} + \Frac{3s+5}{s^2+10s+50}\right](t)}
\end{enumerate}
%
\newpage
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~\\*[1.5em]
%
\item Find the Laplace Transform of the solutions, 
\eqline{X(s)=\LC[x(t)]} and \eqline{Y(s)=\LC[y(t)]}, 
satisfying the system of ODEs,
\eqlg{x'(t) + 2y(t) = 8e^{\DSB t}\sin(t)\delta(t-5), ~~ x(0) = 6,
\\*[1ex]
x(t) - y'(t) = 7u(t-5), ~~ y(0)=0.\hspace*{0.3in}~}
(Solve, but do not simplify.)
%
\item Some partial PDE problems:
\begin{enumerate}
%
\item Find the \underline{steady state solution} \underline{\underline{only}}
for the inhomogeneous PDE (heat equation) problem,
\eqlg{\mbox{\bf PDE:} ~~ \frac{\partial u}{\partial t}(x,t)
= 5 \frac{\partial^2 u}{\partial x^2}(x,t) - 10 e^{\DSB x}, 
~~ 0 < x < 7, ~~ t > 0;
\\*[1ex]
\mbox{\bf BC:} ~~ u(0,t) = 20, ~~ u(7,t) = 80, ~~ t > 0,\hspace*{0.45in}~}
\item Apply the \underline{method of separation of variables} to the PDE
(``Wave Equation''),
\eqlg{\mbox{\bf PDE:} ~~ \frac{\partial^2 u}{\partial t^2}(x,t) 
= 5 \frac{\partial^2 u}{\partial x^2}(x,t), ~~ 0 < x < 7, ~~ t > 0;
\\*[1ex]
\mbox{\bf BC:} ~~ u(0,t) = 0 = u(7,t), ~~ t > 0,\hspace*{0.45in}~}%
and obtain the two ODEs that the function of \eqline{x} and function of
\eqline{t} must satisfy.  Also determine the BC that the function of
\eqline{x} must satisfy (do not solve).
\end{enumerate}
%
\item 
\begin{enumerate}
\item Find the \underline{Fourier series} for the function,
\eqlg{f(x) = \left\{\begin{array}{lc}
0, & -\pi < x < -\pi/2\\
3, & -\pi/2 < x < +\pi/2\\
0, & +\pi/2 < x < +\pi
\end{array}\right\},}%
by computing the Fourier coefficients.
\item Graph the extended function to which the Fourier series converges in 
the interval \eqline{[-2\pi, +2\pi]} and be sure to include the values 
at the points of discontinuity.
\end{enumerate}
\end{enumerate}
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}% end large
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