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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 1999}
\\
{\bf Final Exam}
\\
{\bf Friday, May 7}
\end{center}
%\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm} Examination contains 11 problems on 12 sheets
of paper including the front cover.
Do all your work on the paper provided and show your computations.
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      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  3      &  20  &       \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
  8      &  15 &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  9      &   20 &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  10     &   20 &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  11     &   20 &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline\hline

Total  & 175\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 Suppose that $y(t)$ solves the initial value problem 
$$
y' = \sqrt{x-y}, \qquad y(0) = -2.
$$
Use Euler's method with a single step to approximate $y(.1)$.  Then give
an upper bound for the difference between your approximation and the
actual value of $y(.1)$. (15 points)
\vspace*{4cm}

\newpage

\item
Consider the inhomogeneous differential equation.
$$
y'' - 3y' + 2y = 4\sin 2x
$$
\begin{subproblem}
\item Find the general solution of the corresponding {\it homogeneous} 
equation.
(5 points)
\vspace*{4cm}
\item Find the general solution of the (inhomogeneous) equation.
(10 points)
\vspace{11cm}
\item Convert the inhomogeneous equation into an equivalent $2\times 2$ system 
of the form \break
$
\left(
\begin{array}{c}
 y_1 \\
 y_2  
\end{array}
\right)'
=
\left(
\begin{array}{ccc}
a_{11} & a_{12} \\
a_{21} & a_{22} 
\end{array}
\right)
\left(
\begin{array}{c}
 y_1 \\
 y_2 
\end{array}
\right)
+
\left(
\begin{array}{c}
 g_1 \\
 g_2 
\end{array}
\right).
$
(5 points)
\end{subproblem}

\newpage

\item Suppose that $y(x)$ solves the initial value problem
$$
y'' + y' + xy = 0, \qquad y(0) = 2, y'(0) = 0.
$$
Using any method you like,
find the first three non-zero terms in the power series for $y$
centered at $x=0$.  (15 points)
\newpage

\item Consider the system of ODE's
$$
\left(
\begin{array}{c}
 x_1 \\
 x_2  
\end{array}
\right)'
=
\left(
\begin{array}{cc}
0 & 4 \\
-1 & 4 
\end{array}
\right)
\left(
\begin{array}{c}
 x_1 \\
 x_2 
\end{array}
\right).
$$
\begin{subproblem}
\item Find the general solution of the system. (10 points)
\vspace*{15cm}
\item Use your answer above to write down a fundamental matrix for the
system.  If you failed to find the answer to the first part, make
one up for illustration's sake and get a fundamental matrix from that.
{\bf DO NOT} attempt to find the fundamental matrix by exponentiating!
(5 points)
\end{subproblem} 
\newpage

\item Suppose that a certain system 
$
\mathbf{x'}(t) = A\mathbf{x}(t),
$
of ODE's has a fundamental matrix 
$$
\Psi(t) = 
\left[
\begin{array}{ccc}
 2e^{3t} & e^{-t} \\
 -3e^{3t} & -2e^{-t} 
\end{array}
\right].
$$

\begin{subproblem}
\item
Write down the general solution of
$$
\mathbf{x'}(t) = A\mathbf{x}(t) + 
\left[
\begin{array}{c}
\sin t \\
\cos t 
\end{array}
\right].
$$
{\bf DO NOT} evaluate any integrals or multiply any matrices that appear in 
your answer. (5 points)
\vspace*{6cm}
\item Write down the solution of the corresponding homogeneous equation.
\end{subproblem} (5 points)
\newpage



\item Consider (but {\bf do not try to solve}) the differential equation
$$
y' = (y^2 - 1)(y+1)^2.
$$
\begin{subproblem}
\item Find and classify all equilibrium solutions. (10 points)
\vspace*{8cm}

\item On a single plot, sketch the graphs of the solutions $y_1,y_2,y_3$ 
satisfying $y_1(0) = 0$, $y_2(0) = 1$, $y_3(0) = 2$.  Make sure label 
to label the graphs! (5 points)
\end{subproblem}
\newpage

\item 
Find the critical points of the non-linear system of ODE's
\begin{eqnarray*}
x' & = & (x-2)(y+1) \\
y' & = & x-2y
\end{eqnarray*}
For one of these critical points (take your pick...), find the corresponding
linear system and determine whether the critical point is stable
or unstable. (15 points)
\newpage

\item Existence and uniqueness for first order ODE's.
\begin{subproblem} 
\item
State the existence and uniqueness theorem for an initial value problem
of the form
$$
\frac{dy}{dt} = f(y,t), \qquad y(t_0) = y_0.
$$
(5 points)
\vspace*{4cm}

\item Give an example of such a problem in which one of the hypotheses of
the existence and uniqueness theorem fail. (5 points)
\vspace*{4cm}

\item How does the conclusion of the theorem improve if 
$f(x,t) = -p(t)y + q(t)$? (5 points)
\vspace*{4cm}

\item Intuitively speaking, what does the existence and uniqueness theorem
say if we start plotting solutions $y(t)$ of the ODE. (5 points)

\end{subproblem}
\newpage

\item 
Consider the following list of words that we used in class to 
classify ODEs:
\begin{description}
\item{A.} autonomous,
\item{B.} constant coefficient,
\item{D.} homogenous,
\item{E.} linear.
\end{description}

For each of the ODE's below, indicate all of the words on the
list that accurately describe the ODE. (5 points each)

\begin{subproblem}
\item $\displaystyle\frac{dy}{dx} + xy = 3$
\item $y'' - y^2 = 2$
\item $y'''(t) = 3y'(t) + 2y(t)$
\item $\left(\displaystyle\frac{d^2y}{dt^2}\right)^2 - x^2 y 
            - 2\sin x = 0.$
\end{subproblem}

\item Answer the following questions concerning direction fields and
      phase portraits. (5 points each)

\begin{subproblem}

\item The following direction field comes from a first order, linear
      homogeneous $2\times 2$ system $\mathbf{x}' = A\mathbf{x}$.
      Draw in the eigenvectors for $A$ and say what you can about the
      corresponding eigenvalues.

\epsfig{file=final_fig4.eps,height=8cm,width=8cm}
\newpage

\item Which of the following direction fields comes from a first
      order autonomous ODE $y' = f(y)$.  Why?

\epsfig{file=final_fig1.eps,height=5cm,width=5cm}
\epsfig{file=final_fig2.eps,height=5cm,width=5cm}
\epsfig{file=final_fig3.eps,height=5cm,width=5cm}
\vspace*{3cm}

\item The following direction field comes from a first order, non-linear
      $2\times 2$ autonomous system of ODE's.  Identify the critical
      points of this system and say whether each is stable or unstable.

\epsfig{file=final_fig5.eps,height=8cm,width=8cm}
\newpage

\item Consider the non-linear system 
      \begin{eqnarray*}
      x' & = & (x-2)(y-1) \\
      y' & = & x-2y
      \end{eqnarray*}
      from problem 8 on this exam.  Note that the point $(x,y) = (1,0)$ is
      {\bf NOT} a critical point of the system.  What do you expect the 
      phase portrait for this system to look like very close to $(1,0)$?  
      You can
      either draw a picture or give a verbal description for your answer.
\end{subproblem}



\end{problems}



\end{document}
\end

