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\begin{center}
  \textbf{Comments on Maple Problem Set F, Problems 5 and 13}
\end{center}

Be sure to read the instructions for the problems carefully and do all
parts.  
You must use Maple on these problems.  Do not use pplane5 to do them.
(Of course, if you want to use pplane5 to help you see what to expect
from Maple, that's OK--but the solutions you hand in cannot use pplane5).

\bigskip


 You can use either {\bf DEplot} (see p.\ 169 of \emph{Differential
  Equations with Maple}) or odeplot to plot numerical solutions of
  systems.  There are probably other ways as well.
  Here is how to do
  this with {\bf odeplot}
 for a system of two equations.  Of course, you have to give
  specific
funtions 
\begin{math}
  f
\end{math}
and 
\begin{math}
  g
\end{math},
values for 
\begin{math}
  m,n
\end{math},
and you may want to adjust the range of 
\begin{math}
  t
\end{math},
and the range of values of 
\begin{math}
  a,b.
\end{math}

\medskip

 $ >$ with(plots):

 $ >$ sys := diff(x(t),t)=f(x(t),y(t)), diff(y(t),t)=g(x(t),y(t));

 $ >$ numsol := (a,b) -$>$
    dsolve(\{sys,x(0)=a,y(0)=b\},\{x(t),y(t)\},numeric);

 $ >$ curve := (a,b,trange) -$>$ odeplot(numsol(a,b),[x(t),y(t)],trange);

 $ >$ nphase := trange -$>$
    display(\{seq(seq(curve(a,b,trange),a=1..n),b=1..m)\});

 $ >$ nphase(0..10);

\medskip

\emph{Problem 5}
\begin{itemize}
\item You may need to use a fairly large number of points (for
  example, {\bf numpoints=1000}) in your plot commands to get
  reasonable plots.
\item In part (a), if you get a plot which cannot be correct, please
indicate that, and also explain what is wrong with the plot.
\item In part (c), you can either differentiate 
  \begin{math}
    E(t)
  \end{math}
by hand or have Maple do it.  If you have Maple do it, be sure
to use 
\begin{math}
  x(t),y(t) 
\end{math}
in your formula for 
\begin{math}
  E(t)
\end{math}
instead of 
\begin{math}
  x,y
\end{math}
since otherwise, Maple will think 
\begin{math}
  E(t)
\end{math}
is a constant function.
\item Once you have a formula for the derivative, you need to 
substitute the formulas for 
\begin{math}
  x',y'
\end{math}
along a trajectory.
\item Parts (b), (f) and (g) require careful thorough explanations.
\end{itemize}

\emph{Problem 13}
\begin{itemize}
\item In (b) if you don't set a large enough range for 
  \begin{math}
    t
  \end{math}
you won't get a very good phase portrait.  On the other hand, if it's
too large, Maple will have problems doing some of the computations.

\item In (b), include both a graphical analysis and a mathematical analysis
  of stability.  
\end{itemize}

\end{document}
