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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 Final Exam}
\\
{\bf Tuesday, May 8}
\end{center}
%\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm} Examination contains ten problems on nine sheets
of paper including the front cover.  Do all your work on the paper provided
and show your computations. {\bf When it makes sense to do so, please box your
answers---this makes it much easier for me to see what you're doing.}

\\[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      &  5  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  3      &  20  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  4      &  30 &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  5      &  10  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  6      &  20  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  7      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  8      &  10  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  9      &  20  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  10      &  10 &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline\hline
Total  & 150\mbox{\  \ } & \rule[-.3cm]{0cm}{0.9cm} \\ \hline
\end{tabular}
\end{center}

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

\begin{center} {...and enjoy your summer!}
\end{center}
\newpage

\begin{problems}
\item Solve the initial value problem
$$
y' +  t y - t = 0, \qquad y(-1) = 2.
$$
(10 points)
\vspace*{5in}

\item Consider {\it but do {\bf NOT} try to solve} the initial value problem
$$
(1-t) y'' + (2-t) y = \ln (16 - t^2), \qquad y(3) = 5, y'(3) = 10^6.
$$
On what interval is this problem guaranteed to have a solution?
(5 points)
\newpage


\item Suppose that $y(t)$ solves the initial value problem 
$$
y' = \sqrt{t+y^2}, \qquad y(0) = 2.
$$
\begin{subproblem}
\item
Use Euler's method with a single step to approximate $y(.1)$.  
(10 points)
\vspace*{2in}
\item Estimate the difference between your approximation and the
actual value of $y(.1)$---that is, estimate the local truncation
error.  You needn't simplify any of your
arithmetic.  
(10 points)
\end{subproblem}
\newpage

\item
Consider the inhomogeneous differential equation.
$$
y'' + 2y' + 2y = \sin 2t + t
$$
\begin{subproblem}
\item Find the general solution of the corresponding {\it homogeneous} 
equation.  (10 points)
\vspace*{7cm}
\item Find the form of the general solution of the (inhomogeneous)
equation, and circle any coefficients that must be determined. 
{\it Do {\bf NOT} try to determine the coefficients.} (10 points)
\vspace{7cm}
\item Convert the inhomogeneous equation into an equivalent $2\times 2$ system 
of first order ODE's, and write the system in matrix form. 
(10 points)
\end{subproblem}

\newpage

\item Find the general solution of the system 
$$
\left(
\begin{array}{c}
 x_1 \\
 x_2  
\end{array}
\right)'
=
\left(
\begin{array}{cc}
3 & -2 \\
2 & -1 
\end{array}
\right)
\left(
\begin{array}{c}
 x_1 \\
 x_2 
\end{array}
\right).
$$
(10 points)
\newpage


\item Consider (but do {\bf NOT} try to solve) the autonomous differential
equation 
$$
y' = (y - 10)(y - 20)(y-30)^2.
$$
\begin{subproblem}
\item Find and classify the critical points of this equation.
(10 points)
\vspace*{3.5in}
\item Given a number $0\leq y_0\leq 20$ and a solution $y(t)$ with initial
      value $y(0) = y_0$, what happens to the value of $y(t)$ as $t$ increases?
      Note that the answer might depend on $y_0$.
(10 points)
\vspace*{2in}
\end{subproblem}
\newpage


\item Suppose that $A$ is a $2\times 2$ constant matrix and that the direction
field for $\mathbf{x}' = A\mathbf{x}$ is as follows:

\begin{center}
\epsfig{file=finfig1.eps,height=5cm,width=5cm}
\end{center}
\begin{subproblem}
\item
Highlight any visible eigenspaces in the direction field.
(6 points)
\item
Write down a plausible general solution $\mathbf{x}(t)$ for this system. Of
course your numbers won't be exactly right---close is good enough. 
(9 points)
\vspace*{2cm}
\end{subproblem}

\item Here is a phase portrait/direction field for a certain $2\times 2$
(non-linear) autonomous system of ODE's.
\begin{center}
\epsfig{file=finfig2.eps,height=8cm,width=8cm}
\end{center}
Identify each critical point of the system along with its type and
stability.
(10 points)
\newpage


\item Consider the non-linear autonomous system 
\begin{eqnarray*}
x' & = & (x-y)(y-2)\\
y' & = & xy-1
\end{eqnarray*}.
\begin{subproblem}
\item Find the critical points of this system.
(10 points)
\vspace*{2in}

\item For one of the critical points (your choice) write down the
corresponding linear system and draw conclusions (if any) about the
stability of that point.
(10 points)
\end{subproblem}
\newpage


\item Having become expert at solving systems of ODE's, you amuse yourself
by finding the general solution to a linear, homogeneous $2\times 2$ system of 
ODE's specified in a random homework problem from chapter 7 in Boyce and
DiPrima. You obtain
$$
\mathbf{x}(t) = 
C_1 e^t \left(
\begin{array}{c}
 2\cos 3t - \sin 3t \\
 \cos 3t  
\end{array}
\right)
+
C_1 e^t \left(
\begin{array}{c}
 \cos 3t + 2\sin 3t \\
 \sin 3t  
\end{array}
\right).
$$
There is, of course, no chance that you have made a mistake.  However, out of
concern for your fellow, more fallible classmates, you compare your solution
with the answer given in the back of the book to make sure that the book is
right.  The book says: $$
\mathbf{x}(t) = 
A e^t \left(
\begin{array}{c}
 \cos 3t \\
 \cos 3t - \sin 3t  
\end{array}
\right)
+
B e^t \left(
\begin{array}{c}
  \sin 3t \\
 \cos 3t + \sin 3t  
\end{array}
\right).
$$
Is the book right?  Why or why not?
(10 points)
\end{problems}


\end{document}
\end



