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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 2}
\\
{\bf Friday, April 20}
\end{center}
%\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm} Examination contains 5 problems on 7 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      &  25  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  2      &  20  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  3      &  30  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  4      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  5      &  10  &       \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 following inhomogeneous differential equation
$$
y''' + 2y'' + y' = \cos t - 2e^{-t}.
$$
\begin{subproblem}
\item  What form will the general solution of this equation have?  
Circle the arbitrary constants that must be determined, but do {\bf NOT}
try to actually solve for them.  (15 points)
\vspace*{4in}
\item  Rewrite this equation as a first order system of equations, and 
       express the system in matrix form.  (10 points)
\end{subproblem}
\newpage


\item 
Consider the system
$$
\left(
\begin{array}{c}
 x_1 \\
 x_2  
\end{array}
\right)'
=
\left(
\begin{array}{cc}
4 & -1 \\
1 & 4 
\end{array}
\right)
\left(
\begin{array}{c}
 x_1 \\
 x_2 
\end{array}
\right).
$$
\begin{subproblem}
\item
Find the general solution. (15 points)
\newpage

\item
What further equations must you solve to find the particular solution
satisfying  $$
\left(
\begin{array}{c}
 x_1(\frac{\pi}{2}) \\
 x_2(\frac{\pi}{2}) 
\end{array}
\right)
=
\left(
\begin{array}{c}
 \pi \\
 e^2
\end{array}
\right)?
$$
You need not actually solve the equations, but I do expect you
to know the values of trig functions at basic angles.  (5 points)
\vspace*{2in}

\ignore{
\itemConvert this first order 2x2 system into a single second order ODE
}

\end{subproblem}




\item Suppose that $\mathbf{x}' = A\mathbf{x}$ is a $2\times 2$ system of 
ODE's, and that $A$ is a constant matrix.  Suppose that
$$
\mathbf{x}^1(t) = \left(\begin{array}{c} 1 \\ 2 \end{array} \right) e^{2t}
           \qquad{\text{and}}\qquad 
\mathbf{x}^2(t) =  \left(\begin{array}{c} -2 \\ 3 \end{array}\right) e^{3t} 
$$
are solutions to this system.

\begin{subproblem}
\item What is $e^{At}$?  You don't need to simplify your answer to this one.
(5 points)
\vspace*{1.5in}


\item What sort of critical point is $(0,0)$ for this system?  Is it stable
      or unstable?  
(5 points)
\newpage

\item On the graph below, plot solution curves for solutions $\mathbf{x}(t)$
      satisfying $\mathbf{x}(0) = (1/2,1)$ and $\mathbf{x}(0) = (0,-1)$.
      Draw arrows on your solution curves to indicate the direction for
      increasing $t$.
      (10 points)
\begin{center}
\epsfig{file=ex2fig1.eps,height=10cm,width=10cm}
\end{center}
\item Suppose that we change the system to
$$
\mathbf{x}' = A\mathbf{x} + 
    \left(\begin{array}{c} t^2 \\ t^3 \end{array} \right)
$$
where $A$ is the same matrix as before.  Write down the solution to 
this new system.  {\it Do {\bf not} simplify your answer---i.e. don't perform
any matrix multiplication, inversion, integration, differentiation, etc.  (10
points)}   \end{subproblem}
\vspace*{2in}


\ignore{\item Suppose that $\mathbf{x^1}$ and $\mathbf{x^2}$ are both solutions
of a first order linear system
$$
\frac{d\mathbf{x}}{dt} = A\mathbf{x}.
$$
Show that any linear combination $\mathbf{x}(t) = C_1\mathbf{x}^1 +
C_2\mathbf{x}^2$ (where $C_1, C_2$ are constants) is also a solution.} 
\newpage


\item Each of the following three pictures represents a direction 
field or phase portrait for a system of ODE's of the form 
$$
\mathbf{x}' = A\mathbf{x},
$$
where $\mathbf{x}(t) = (x_1(t), x_2(t))$ is a vector-valued function and 
$A$ is $2\times 2$ constant matrix.  For each picture,
say what can be said about the eigenvalues and eigenvectors of the matrix $A$.
(5 points each) \vspace*{1cm}  

\begin{description}
\item{(a)}
\epsfig{file=ex2fig2.eps,height=5cm,width=5cm}
\vspace*{1cm}
\item{(b)}
\epsfig{file=ex2fig3.eps,height=5cm,width=5cm}
\vspace*{1cm}
\item{(c)}
\epsfig{file=ex2fig4.eps,height=5cm,width=5cm}
\end{description}
\newpage

\item
Suppose that $A$ is a $2\times 2$ constant matrix and that $B = 2A$.  Compare
solutions of the system 
$$
\mathbf{x}' = A\mathbf{x}
$$
with those of 
$$
\mathbf{x}' = B\mathbf{x}.
$$
For instance, do the direction fields and phase portraits of the two
systems have anything in common?  How about the actual formulas for the
general solutions?  Try to justify your responses.  It might also help
to consider a particular example.
(10 points)






\end{problems}




\end{document}
\end

