%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[12pt]{article}
\usepackage{amssymb}
\usepackage{epsfig}
\topmargin-2cm
\textheight 24cm
\textwidth18cm
\oddsidemargin-.7cm
\evensidemargin-.7cm
\parindent0cm
\parskip 2.5ex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\re}{\rm Re}
\newcommand{\im}{\rm Im}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Ds}{\displaystyle}
\newcommand{\dfrac}[2]{{\Ds \frac{#1}{#2}}}
\newcommand{\junk}[1]{}
\newcommand{\text}[1]{\mbox{\rm{#1}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcounter{prob}
\newcounter{subprob}
\newenvironment{problems}{
     \begin{list}{\bf{\arabic{prob}.}\hfill}{\usecounter{prob}
     \topsep-1.4ex \labelwidth.6cm \leftmargin.7cm \labelsep.1cm
     \rightmargin0cm \parsep2ex \itemsep2ex
     \partopsep1.4ex}}{\end{list}}
\newenvironment{subproblem}{
     \begin{list}{\bf{(\alph{subprob})}\hfill}{\usecounter{subprob}
     \topsep-1.4ex \labelwidth.6cm \leftmargin.8cm \labelsep.2cm
     \rightmargin0cm \parsep2ex \itemsep2ex
     \partopsep1.4ex}}{\end{list}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}

{\bf Name:} \underline{\mbox{\hspace*{7cm}}}
\hfill{\bf Instructor:} Jeffrey Diller

\begin{center}
{\bf Math 230: Ordinary Differential Equations }
\\
{\bf Spring Semester 1999}
\\
{\bf Exam 2}
\\
{\bf Thursday, April 15}
\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.
Do all your work on the paper provided and show your computations.
\\[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      &  25  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  2      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  3      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  4      &  10  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  5      &  10  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  6      &  10  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  7      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline


Total  & 100\mbox{\  \ } & \rule[-.3cm]{0cm}{0.9cm} \\ \hline
\end{tabular}
\end{center}

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

\begin{problems}

\item Consider the inhomogeneous differential equation.
$$
y'' + 4y' + 4y = e^{2t} + e^{-2t}
$$
\begin{subproblem}
\item Find the general solution of the corresponding {\it homogeneous} 
equation.
(10 points)
\vspace*{5cm}
\item Find the general solution of the (inhomogeneous) equation.
(10 points)
\newpage
\item Find an equivalent first order system of differential equations,
and express this system using matrix/vector notation. (5 points)
\end{subproblem}
\vspace*{4cm} 







\item 
Consider the system
$$
\left(
\begin{array}{c}
 x_1 \\
 x_2  
\end{array}
\right)'
=
\left(
\begin{array}{cc}
0 & -3 \\
3 & 0 
\end{array}
\right)
\left(
\begin{array}{c}
 x_1 \\
 x_2 
\end{array}
\right).
$$
\begin{subproblem}
\item
Present the solution in real form (i.e. without imaginary numbers).
(10 points)
\vspace*{8cm}
\item
Find the particular solution satisfying 
$$
\left(
\begin{array}{c}
 x_1(0) \\
 x_2(0) 
\end{array}
\right)
=
\left(
\begin{array}{c}
 1 \\
 0 
\end{array}
\right)
$$
(5 points)
\end{subproblem}
\newpage




\item 
Let $\sum_{n=0}^\infty c_n x^n$ be a power series solving
$$
y'' + (x+1) y = 0, \qquad y(0) = 1, y'(0) = 0.
$$
Find a recursion formula for the coefficients $c_n$ and give the
first four non-vanishing terms in the series.
(15 points)

\newpage





\item Consider the functions $e^x, x+1$ on the interval 
$(-\frac12,\frac12)$.
\begin{subproblem}
\item Are these functions linearly independent? Why or why not? (5 points)
\vspace*{4cm}
\item Could there be a differential equation $y''(x) + p(x)y'(x) + q(x) = 0$
having both these functions as solutions?  Why or Why not? 
Assume $p(x)$ and $q(x)$ are continuous for $x\in(-\frac12,\frac12)$. 
(5 points)
\end{subproblem}
\vspace{4cm}


\item Suppose that $\sum_{n=0}^\infty c_n x^n$ solves the initial 
value problem
$$
y'' + \frac{1}{x^2 + 2} y = 0, \qquad y(0) = 0, y'(0) = 1.
$$
\begin{subproblem}
\item What is $c_3$? (Hint: do {\bf not} try to find a recursion formula
      for the coefficients $c_n$.) (5 points)
\vspace*{4cm}
\item What is the (smallest possible) radius of convergence of the series?
(5 points)
\end{subproblem}
\newpage


\item 
Find and classify the singular points of 
$$
(e^x - 1)(x-1)^2 y'' + xy' + \frac{x-1}{x} y = 0.
$$
(10 points)
\newpage




\item Each of the following three pictures represents a direction 
field or phase portrait for a $2\times 2$ linear, homogeneous,
constant coefficient system of ODE's---that is, a system of the form 
$$
\mathbf{x}'(t) = A\mathbf{x}(t),
$$
where $\mathbf{x(t)} = (x_1(t), x_2(t))$ is a vector-valued function and 
$A$ is $2\times 2$ constant matrix.  Note that $A$ is different from
one picture to the next.

For each picture, say what you can 
about the eigenvalues and eigenvectors of the matrix $A$ in the 
corresponding system. (5 points each)
\vspace*{1cm}

\begin{description}
\item{(a)}
\epsfig{file=exam2_fig1.eps,height=6cm,width=6cm}
\vspace*{2cm}
\item{(b)}
\epsfig{file=exam2_fig2.eps,height=6cm,width=6cm}
\item{(c)}
\epsfig{file=exam2_fig3.eps,height=6cm,width=6cm}
\end{description}

\begin{center}

\end{center}






\end{problems}



\end{document}
\end
