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\begin{document}
{\bf Name:} \underline{\mbox{\hspace*{7cm}}}
\hfill{\bf Instructor:} Heide Gluesing-Luerssen

\begin{center}
{\bf Math 226: Differential Equations and Linear Algebra }
\\
{\bf Fall Semester 1998}
\\
{\bf Exam 2}
\\
{\bf November 9, 1998}
\end{center}
%\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm} Examination contains 7 problems on 8 sheets
of paper including the front cover (not counting 2 blank pages at the end).
Do all your work in this booklet and show your computations.
Calculators, books, and notes are not allowed.
\\[1.5ex]
\hline
\end{tabular}

\begin{center} {\bf Scores}

\begin{tabular}{|c|c|c|}\hline
Question & Possible\quad &\ Actual\quad\rule[-.3cm]{0cm}{0.9cm} \\ \hline\hline
  1      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  2      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  3      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  4      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  5      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  6      &  10  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  7      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline\hline
Total  & 100\mbox{\  \ } & \rule[-.3cm]{0cm}{0.9cm} \\ \hline
\end{tabular}
\end{center}

\vspace*{1cm}
{\bf Sign the pledge:}
\\
``On my honor, I have neither given nor received unauthorized aid on this
Exam.''

{\bf Signature:} \underline{\mbox{\hspace*{7cm}}}

\vspace*{.5cm}
\begin{center}     {\bf GOOD LUCK}\end{center}




\newpage
\begin{problems}




\item
Solve the following initial value problem.
\[
    y'' + 4y' + 5y = 0, \quad y(0) = 1, \quad y'(0) = 1.
\]
Your answer should be in real form (i.e. no complex numbers). (15 points)
\vspace*{20cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}



\item
Find the general solution $y(x)$ of 
\[
y'' - 4y' + 4y = 3e^{-3x}.
\]
(15 points)

\vspace*{21cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}







\item
Consider the following three pairs of functions.
\begin{enumerate}
\item[A.] $3\sin x$, $4\cos x$;
\item[B.] $\sin x$, $2\sin x$;
\item[C.] $x^2$, $3x^4$.
\end{enumerate}

\begin{subproblem}
\item Compute the Wronskian of each pair. (5 points)
\vspace*{6cm}
\item Which of the pairs are linearly independent?  Explain briefly.
(5 points) 
\vspace*{5cm}
\item Which pairs consist of functions that could not both solve the same 
differential equation 
$$
y'' + p(x)y' + q(x)y = 0, \quad -\infty< x < \infty
$$ 
(assume $p$ and $q$ are continuous everywhere)?  Explain 
briefly. 
(5 points)
\vspace*{5cm}
\end{subproblem}





\item
The function $y(x) = 1/x$ solves the differential equation
$$
xy'' + (2-2x)y' -2y = 0.
$$
Find the general solution to the equation. (15 points)
\vspace*{21cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}


\item
Consider the solution $y(x)$ of the initial value problem  
$$
y'' + 2x y = 0, \quad  y(0) = 2, \quad y'(0) = 3.
$$
Let $\sum_{n=0}^\infty a_n x^n$ be a power series representing
$y$.
\begin{subproblem}
\item Write down a recurrence relation for finding the coefficients 
      $a_n$. (10 points)

\vspace{13cm}
\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}
\item Compute the values of the coefficients $a_0, a_1, a_2, a_3$.
      (5 points)

\vspace{4cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}
\end{subproblem}





\item
The equation
$$
(x^2 - 3x) y'' + x y' + (x-2) y = 0
$$
has a power series solution $y = \sum_{n=0}^\infty a_n(x-2)^n$.
What is the largest interval on which this series is certain to
converge? Make sure your work explains your answer.  (10 points)
\vspace*{20cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}

\item
Suppose that you are designing the suspension system for a new car.
The 1000 kilogram vehicle is mounted on its springs without any
shock absorbers (i.e. without damping).  As a test, you push the
car down and let it bounce back up.  It bobs up and down about
once a second.  

\begin{subproblem}
\item Write down a differential equation that describes the motion
      of this undamped system.  Compute the values of
      the coefficients occurring in the equation.  (10 points)

\vspace*{10cm}
\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}

\item What sort of damping would the shock absorbers need to 
      provide (i.e. what would the damping constant need to be) 
      in order to make this system critically damped?  (5 points)

\vspace{7cm}
\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}

\end{subproblem}



\end{problems}
\end{document}