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\begin{document}
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{\large
\begin{center}
{\bf MATH 220 E1 \hspace{0.4in} EXAM 2 \hspace{0.4in} 
12pm 09 April 1999 \hspace{0.4in} Hanson}\\
Laplace Transform and Power Series Solutions of ODEs
\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.
{\bfseries Keep your eyes on your own work and keep your own work covered.}
See {\it Table of Laplace Transforms} on Back.
\\{\centering\underline{~\hspace*{6.85in}~}}
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\begin{enumerate}
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\item
\begin{enumerate}
\item $\DSB \LC\left[7 e^{-8t}\cos(3t)+ t^5 + 11 t^2 \delta(t-3)\right]$
\item $\DSB \LCI\left[\frac{e^{-2s}}{s^2+12s+32}\right]$ 
\hfill(25 points total)
\item $\DSB \LCI\left[\frac{s}{s^2+6s+13}\right]$
\end{enumerate}
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\item
\begin{enumerate}
\item Find the general solution $y(x)$: 
$$\DSB 4 x^2 y^{\DSB\prime\prime}(x) + 8 x y^{\DSB\prime}(x)+ y(x) = 0, 
~~ x>0.$$
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\item Find $z(t)$:  \hfill(25 points total)
$$\DSB z^{\DSB\prime\prime}(t) + 3 z^{\DSB\prime}(t) + 2 z(t) 
= 4\delta(t-\epsilon), 
~~ z(0)=0, ~~ z^{\DSB\prime}(0)=5, ~~ \epsilon>0.$$
What happens as $\epsilon \longrightarrow 0^+$?
\end{enumerate}
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\item 
\begin{enumerate}
\item Find a power series approximation to the solution $y(x)$ of the IVP: 
$$\DSB y^{\DSB\prime\prime}(x) + 2 x y^{\DSB\prime}(x) - 4 y(x)=0, 
~~ y(0)=2, ~~ y^{\DSB\prime}(0)=1,$$
by assuming an expansion about $x = 0$ of the form 
$\DSB y(x)\simeq a_0+a_1 x + a_2 x^2 +a _3 x^3 + a_4 x^4 +a_5 x^5$,
finding the coefficients \{$a_0,a_1,a_2,a _3,a_4,a_5$\}.
You do not need any more terms!
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\item Classify all finite non-negative points of the ODE:
\hfill(25 points total)
$$\DSB x^2 y^{\DSB\prime\prime}(x) + x y^{\DSB\prime}(x) +(x^2-1/9) y(x) = 0,
~~ x > 0$$
and then give only the general form of the power series solution in 
the regular case (say about $x=a$)
and the singular case, but do not evaluate any coefficients.  
\end{enumerate}
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\item Solve the integral equation for $y(t)$:
\hfill(25 points total)
$$\DSB y(t) + 8 \int_0^t y(v)\sin(t-v)dv = 9, ~~ t \geq 0.$$
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\end{enumerate}
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\vfill
{\centering\underline{~\hspace*{6.85in}~}}\\
Math 220 E1 Exam 2  End\hspace*{1.4in}a\hfill(100 points grand total)
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\newpage
{\large\bf
\begin{center}
{\bf Table of Laplace Transforms}\footnote{From Back Cover of Nagle and Saff
Text}
\end{center}
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\begin{center}
\begin{tabular}{||l||l||}\hline\hline
$\DSB f(t)=\LCI(s)](t)$ & $\DSB F(s)=\LC[f(t)](s)$ \\ \hline \hline
$\DSB f(at)$ & $\DSB \frac{1}{a} F\left(\frac{1}{a}\right) $\\
$\DSB e^{at}f(t)$ & $\DSB F(s-a)$ \\
% $\DSB f^{\DSB\prime}(t)$ & $\DSB  sF(s) - f(0)$ \\
$\DSB f^{(n)}(t)$ & $\DSB s^nF(s) - s^{n-1}f(0)- \cdots - f^{(n-1)}(0)$ \\
$\DSB t^n f(t)$ & $\DSB (-1)^n F^{(n)}(s)$ \\
$ \DSB \frac{1}{t} f(t), f(0)=0$ & $\DSB \int_s^\infty F(u) du$ \\
$\DSB \int_0^t f(v) dv $ & $\DSB F(s)/s$ \\
$\DSB  (f * g)(t) $ & $\DSB F(s)\cdot G(s)$ \\
$\DSB  f(t-a)u(t-a), \ a \ge 0$ & $\DSB e^{-as} F(s)$ \\
$\DSB  g(t)u(t-a), \ a \ge 0$ & $\DSB e^{-as} \LC[g(t+a)](s)$ \\
$\DSB  e^{at} \sin(bt)$ & $ \DSB \frac{b}{(s-a)^2 + b^2}$ \\
$\DSB  e^{at} \cos(bt)$ & $ \DSB \frac{s-a}{(s-a)^2 + b^2}$ \\
$\DSB  \sinh(bt)$ & $ \DSB \frac{b}{s^2 - b^2}$ \\
$\DSB  \cosh(bt)$ & $ \DSB \frac{s}{s^2 - b^2}$ \\\hline\hline
\end{tabular}
\end{center}
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