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\noindent
Math 325

\noindent
Spring, 2001

\bigskip

\textbf{Reminder}:  The midterm is Wednesday, March 7.  It will
cover everything we have done so far through Chapter 7, so Chapter  4, 
Sections 8.1, 8.3, 8.5,
 Chapter 7, and Sections 9.1-9.2 in Boyce and DiPrima and   
Chapters 5, 7, 8, and 12
of {\it Differential Equations with Maple}.
You may bring a summary (one side of an  $8\frac 12'' \times 11''\/$
sheet of paper, with notes in your writing) to the midterm.

\bigskip
On the course web page, under \emph{Information}
 you will find a \emph{Review for the
Midterm}, which is an outline of the major ideas and topics we have covered.  
You will also find \emph{Tips for Studying for the Midterm}, which will
give you some idea of what to expect on the exam.

The first project will be due Wednesday, April 4.  Please be
sure to read the guidelines for group projects.  The guidelines
and the project are posted on the course web site under 
\emph{Projects}.

\bigskip

\begin{center}
{\bf ASSIGNMENT 6, due Friday, March 23}
\end{center}

\bigskip


Read Boyce and DiPrima, Sections 9.2-9.3.

Do p. 478 \#5,24, p. 487 \#10 and the \emph{Gallery of Pictures} below. 




\begin{center}
  \textbf{Gallery of Pictures}
\end{center}

This assignment will give you a good feel for the \emph{qualitative}
behavior of constant coefficient systems of two ODEs.  
Use pplane5 to do the plots, and use the Matlab program \textbf{multigraf}
to put several plots on the same sheet of paper.  Be sure to label
your plots.  

\begin{enumerate}
\item Print Figures 1-3 or reproduce them using pplane5.
  Draw arrowheads on the trajectories to indicate the 
direction of increasing time.  Highlight the eigenspaces.  The
\begin{math}
  \lambda
\end{math}
\emph{eigenspace}
is a subspace consisting of the origin and all the eigenvectors with 
eigenvalue 
\begin{math}
  \lambda.
\end{math}

\item
Reproduce Figures 4-7 using pplane5.
Draw arrowheads on the trajectories
 to show the direction of increasing time.
Highlight eigenspaces.

\item 
\label{gallery}
For each system below, use pplane5 to draw a phase portrait for 
\begin{math}
  |x|\le 2,\ |y|\le 2.
\end{math}
On the graph, write the eigenvalues of the system matrix, identify any visible eigenspaces and the corresponding eigenvalue, and for (a)-(j), give the
portrait its name from the list in Table 9.1.1 on p.\ 468 of Boyce
and DiPrima and its stability type.  (\emph{Note}: This list has ten types,
since for 
\begin{math}
  r_1=r_2,
\end{math}
proper and improper nodes are different types.  You will use each type 
once.)  Indicate the advance of time by arrowheads on the trajectories.
Highlight eigenspaces.

\newpage

\begin{enumerate}
\item 
  \begin{math}
    x'=-x,\ y'=-2y
  \end{math}

\item 
  \begin{math}
    x'=-x,\ y'=-y
  \end{math}

\item 
  \begin{math}
    x'=-x,\ y'=x-y
  \end{math}

\item 
  \begin{math}
    x'=x, \ y'=2y
  \end{math}

\item
  \begin{math}
    x'=x,\ y'=y
  \end{math}

\item
  \begin{math}
    x'=x,\ y'=x+y
  \end{math}

\item
  \begin{math}
    x'=x,\ y'=-2y
  \end{math}

\item
  \begin{math}
    x'=-x+10y,\ y'=-10x-y
  \end{math}

\item
  \begin{math}
    x'=10y, \ y'=-10x
  \end{math}

\item
  \begin{math}
    x'=x+10y,\ y'=-10x+y
  \end{math}

\item
  \begin{math}
    x'=0,\ y'=0
  \end{math}

\item
  \begin{math}
    x'=0, \ y'=x
  \end{math}

\item
  \begin{math}
    x'=0, \ y'=-y
  \end{math}

\item
  \begin{math}
    x'=0,\ y'=y
  \end{math}
\end{enumerate}

\item 
 For each system in Problem (\ref{gallery}), plot the graphs of the
components and of the solution curve for a ``typical'' trajectory.
(\emph{Note}: You can use the 3d option under graph in pplane5 
to plot the solution curve as well as the components.)

\item
 Systems (k)-(n) in Problem (\ref{gallery}) are degenerate (at least 
one eigenvalue is 0), so they are not listed in Table 9.1.1.  For each 
of these, mark equilibrium points and identify the stability type.


\item  Show that the origin is a saddle point for the system
of the system 
\begin{displaymath}
  x'=x,\ y'=2x-y
\end{displaymath}
and draw a phase portrait of the system.   Explain the striking 
similarity with the plot labeled \emph{Singular First Order Equation}
(Figure 8).

\item
Select a pair of nonorthogonal lines through the origin.  Construct a 
linear system whose matrix has eigenvalues 
\begin{math}
  \lambda _1 = -1,\ \lambda _2 =-2
\end{math}
with these lines as corresponding eigenspaces.  Plot trajectories of your 
system between the lines.  What is the ``name'' of the system (from 
Table 9.1.1).

\end{enumerate}

\newpage
\begin{figure}[t]
\begin{center}
\resizebox{4.5in}{!}{\includegraphics{hw6-fig1.eps}}
%\includegraphics{hw6-fig1.eps}
\end{center}
\caption{Asymptotically stable proper node}
\end{figure}

\begin{figure}[b]
  \begin{center}
   \resizebox{4.5in}{!}{\includegraphics{hw6-fig2.eps}}
  \end{center}
\caption{Asymptotically stable improper node}
\end{figure}

\begin{figure}[t]
  \begin{center}
   \resizebox{4.5in}{!}{\includegraphics{hw6-fig3.eps}}
  \end{center}
\caption{Asymptotically unstable saddle point}
\end{figure}

\begin{figure}[b]
  \begin{center}
    \resizebox{4.5in}{!}{\includegraphics{hw6-lin1a.eps}} 
  \end{center}
  \caption{Saddle}
\end{figure}

\begin{figure}[t]
  \begin{center}
    \resizebox{4.5in}{!}{\includegraphics{hw6-lin2a.eps}} 
  \end{center}
  \caption{Node}
\end{figure}

\begin{figure}[b]
  \begin{center}
    \resizebox{4.5in}{!}{\includegraphics{hw6-lin3a.eps}} 
  \end{center}
  \caption{Center}
\end{figure}

\begin{figure}[t]
  \begin{center}
    \resizebox{4.5in}{!}{\includegraphics{hw6-lin4a.eps}} 
  \end{center}
  \caption{Spiral}
\end{figure}

\begin{figure}[b]
  \begin{center}
    \resizebox{4.5in}{!}{\includegraphics{hw6-singular.eps}} 
  \end{center}
  \caption{Singular First Order Equation}
\end{figure}

\end{document}

