%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 1}
\\
{\bf Thursday, February 18}
\end{center}
%\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm} Examination contains 9 problems on 7 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      &  10  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  2      &  10  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  3      &  10  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  4      &   {}\ 5 &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  5      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  6      &  10  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  7      &  10  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  8      &   15 &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  9      &   20 &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline\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 Solve the initial value problem
$$
ty' - 3y = (t+1), \qquad y(1) = 2.
$$
(10 points)
\vspace*{8cm}

\item Find a canonical set of solutions for
$$
y'' + 7y' - 8y = 0
$$
at the initial point $t_0 = 0$.
(10 points)
\newpage


\item Consider the initial value problem 
$$
y' = 2e^{3yt}, \qquad y(0) = 1.
$$
\begin{subproblem}
\item
Use Euler's method with a single step to approximate $y(.1)$.  (8 points)
\vspace*{4cm}
\item
Estimate the difference between your approximation and the 
actual value of $y(.1)$ (7 points)
\end{subproblem}
\vspace*{8cm}

\item 
Suppose that I use the Runge-Kutta method with a step size of 
$h=.25$ to approximate $y(1) \approx 1.3345$, where $y(t)$ is the 
solution of an initial value problem  
$$
y' = f(y,t), \quad y(0) = 1.
$$
Suppose I believe that this approximation is accurate to 3 
decimal places.  About what value should I take for $h$ if I want 
my approximation of $y(1)$ to be accurate to 11 decimal places? (5 points)
\newpage



\item Consider (but {\bf do not try to solve}) the differential equation
$$
y' = y^4-2y^3.
$$
\begin{subproblem}
\item Find and classify all equilibrium solutions. (8 points)
\vspace*{8cm}

\item On a single plot, sketch the graphs of the solutions $y_1,y_2,y_3$ 
satisfying $y_1(0) = 1$, $y_2(0) = 2$, $y_3(0) = 3$.  Make sure label 
to label the graphs! (7 points)
\end{subproblem}
\newpage

\item Note that $y_1(t) = e^t$, $y_2(t) = e^{t-1}$ are both solutions of
$$
y''- y' = 0.
$$
Verify that $\{y_1,y_2\}$ is not a fundamental set 
of solutions by evaluating the Wronskian of $y_1$ and $y_2$ at $t=0$.  
Then give initial values $y_0,y'_0$ such that no linear
combination $y = C_1y_1 + C_2y_2$ satisfies $y(0) = y_0$ and $y'(0) = y'_0$.
(10 points)

\vspace*{9cm}



\item 
Consider (but {\bf do not try to solve}) the initial value problem
$$
y' = \left(y+\frac{1}{t}\right)^{2/3},   \qquad y(t_0) = y_0.
$$
For what initial values $t_0$ and $y_0$ do the hypotheses of the
existence and uniqueness theorem for first order non-linear ODE's
fail?
(10 points)
\newpage


\item 
Succumbing to Detroit's relentless marketing, I give into my irrational
desire to have a sport utility vehicle.  I rush to the nearest Lincoln
dealer and inquire about prices for their Navigator SUV.  The sales
manager tells me that they'll sell me this vehicle for \$45,000 cash.
Noticing the crestfallen look on my face, he puts his arm around my
shoulder and tells me that though he wouldn't normally do this, he'll
let me drive the Navigator away for free---they'll just loan me the
money at 10\% continuously compounded interest, and I can pay the vehicle off
in continuous payments over five years.  Heck, he'll even knock the
asking price down to \$40,000.  At this point my disappointment turns
to anger, I fling off his unwanted and insincere arm and storm out of
the dealership resolved to endure my Dodge Neon a few years longer.  
How much would I have paid if I had accepted the dealer's second offer?
(15 points)
\newpage

\item Below are direction fields for four first order differential
equations.  Identify the direction field corresponding to
\begin{subproblem}
\item a linear homogeneous equation;
\item an autonomous equation;
\item an equation of the form $\displaystyle\frac{dy}{dt} = f(t)$;
\item a non-linear and non-autonomous equation;
\end{subproblem}
(5 points each)
\vspace*{1cm}

\begin{center}
(i) \epsfig{file=exam1_fig1.eps,height=6cm,width=6cm}
\hspace*{2cm}
(ii) \epsfig{file=exam1_fig2.eps,height=6cm,width=6cm}
\end{center}
\vspace*{1cm}

\begin{center}
(iii) \epsfig{file=exam1_fig3.eps,height=6cm,width=6cm}
\hspace*{2cm}
(iv)  \epsfig{file=exam1_fig4.eps,height=6cm,width=6cm}
\end{center}

\begin{center}

\end{center}






\end{problems}



\end{document}
\end

