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\begin{document}

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

\begin{center}
{\bf Math 230: Ordinary Differential Equations }
\\
{\bf Spring Semester 2001}
\\
{\bf Exam 1}
\\
{\bf Friday, March 2}
\end{center}
%\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm} Examination contains 8 problems on 8 sheets
of paper (including the front cover).
Do all your work on the paper provided and show your computations.
Do not use a calculator.
\\[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      &  15  &       \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      &  20  &       \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 Solve the initial value problem
$$
y' + 3y = e^t, \qquad y(1) = 3.
$$
(10 points)
\vspace*{8cm}

\item Find the general solution of
$$
\frac{d^4y}{dt^4} - 4y = 0.
$$
(10 points)
\newpage

\item
Find the general solution of 
$$
2t^2y'' - 3ty' + 2y = 0
$$
given that $t^{1/2}$ is a solution.   (10 points)

\newpage

\item 
Consider the initial value problem 
$$
y' = \sqrt{y}- t, \qquad y(1) = 4
$$
\begin{subproblem}
\item
Use Euler's method with a single step to approximate $y(1.01)$.  (10 points)
\vspace*{7cm}  
\item
When I use Euler's Method with a step size of $.01$ to estimate $y(2)$ for this
initial value problem, I get an approximate value of $4.593$.  When I drop
the step size to $.001$, the approximation changes to $4.589$.  Based on
this information, about what step size should I choose in order to estimate
$y(2)$ to within $10^{-5}$? 
(5 points)
\end{subproblem}


\newpage



\item Consider the autonomous differential equation
$$
y' = 2y^3(y^2-4).
$$

\begin{subproblem}
\item Identify each equilibrium solution of the equation and classify
it as stable, unstable, or semistable. 
(10 points)
\vspace*{7cm}

\item On a single plot, sketch the graphs of the solutions $y_1,y_2,y_3$ 
satisfying $y_1(0) = 0$, $y_2(0) = -1$, and $y_3(0) = 4$.  Make sure 
to label the graphs! (5 points)
\end{subproblem}
\newpage

\item
\begin{subproblem}
\item Show that the functions $t,t^2,t^3$ are linearly independent. (5 points)
\vspace*{10cm}
\item Suppose that $y''' + p_2(t)y'' + p_1(t)y' + p_0(t)y = 0$ satisfies the
hypotheses of the existence and uniqueness theorem for linear ODE's for
all $t\in\mathbf{R}$.  Explain why $t$, $t^2$, and $t^3$ cannot all be solutions of 
this ODE valid for all $t\in\mathbf{R}$. (5 points)
\end{subproblem}
\newpage

\item 
Bob, Peter, and Paul are roommates living in Bob's house.  Bob owes Paul
\$500 for all the CD's he borrowed and lost last year.  In order
to `facilitate' Bob's  repayment of this debt, Paul begins charging Bob
interest at the rate of $10\%$ per month.  Bob decides that raising Peter's
rent will be the most effective way to pay Paul back.  Peter reacts by
declaring his intention to vacate the premises in 6 months.  Hence Bob needs
your help to figure out how much he'll need to raise Peter's rent in
order to pay Paul back before Peter leaves.  Assuming that the debt accrues 
interest continuously and is paid off continuously, write down
a differential equation and any relevant initial conditions
that describe this situation.  Let Bob find the solution himself.  He's
kind of annoying after all.  
(10 points)

\newpage

\item Here is the direction field for a first order equation differential
equation $y'=f(t,y)$.  Decide whether each of the following statements is
true or false, and {\bf briefly} justify your answer.  (5 points each)

\begin{center}
\epsfig{file=ex1fig1.eps,height=20cm,width=20cm}
\end{center}

\begin{subproblem}
\item The differential equation is autonomous.
\vspace*{2cm}
\item There is a solution $y(t)$ that tends to
$-1$ as $t\to\infty$ and to $\infty$ as $t\to-\infty$.
\vspace*{2cm}
\item Given $t_0 < 0$ and $y_0 < 0$, the solution satisfying $y(t_0)= y_0$ 
satisfies $y(t)<0$ for all $t>t_0$.
\vspace*{2cm}
\item $y=t$ is a solution of the equation.
\end{subproblem}
\end{problems}




\end{document}
\end

