\documentclass[12pt]{article}
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\newcommand{\prob}[3]{\item[\textbf{#1.}] (\textit{expires #2})\quad{#3}}
\newcommand{\M}{{\texttt{Maple}}}
\newcommand{\m}{{\texttt{Maple}}}
\newcommand{\cmd}[1] {{\upshape \color{blue}\texttt{#1}}}
\newcommand{\ds} {\displaystyle}
\newcommand{\R}{{\mathbf{R}}}
\newcommand{\mb}{\mathbf}


\begin{document}

\begin{center}
{\large Math 331, Fall 2002: Problems 13-16}
\end{center}

\textbf{NOTE:} {\itshape Each exercise is worth 10 points and can be
turned in at any time before its ``expiration date''.
At the end of the semester, I will expect you to have
turned in at least 2/5 of the exercises assigned. If you do more, I
will pick your best grades. If you do less, the missing grades will be
counted as zeros. Altogether, these will count the same as one project.
}

\bigskip
%


\begin{enumerate}   

\prob{13}{10/20}{
Find all the solutions to the differential equation
\begin{displaymath}
	\frac{dx}{dt}(t) = -2 x(t), \quad t\in\R.
\end{displaymath}
Among them, single out the one for which $x(0)=3$.
[{\em Hint: read the help page for \cmd{dsolve}, or just do it in your
head. It is that easy.}]
}

\prob{14}{10/20}{
Have \m\ find analytic solutions to the following system of
differential equations,
\begin{displaymath}
	\left\{
	\begin{array}{rcl}
		y''(t) - z(t) &\!=\!& e^t, \\
		z'(t) - y(t) &\!=\!& 0,
	\end{array}
	\right.
\end{displaymath}
with initial conditions: $y(0)=1,\ y'(0)=0,\ z(0)=k$. Let us denote
the solutions by $y_k(t),\ z_k(t)$ (since they depend on the parameter
$k$). 

For $k$ taking all integer values from -10 to 10, and $t \in
[-4,2]$, plot the functions $y_k$ in blue, and the functions $z_k$
in red, all on the same graph. (Yes, you will then have 42 functions
plotted on the same graph.)
[{\em This is certainly a case when you don't want to retype the
functions that \m\ finds.  
You will almost certainly need to read the help page for \cmd{dsolve}.
I also found \cmd{subs}, \cmd{unapply}, and \cmd{seq} useful.}]
}

\prob{15}{10/20}{
For the functions $y_k(t)$ and $z_k(t)$ found in problem \#14, plot
the parametric curves 
$\varphi_k(t) = [y_k(t), z_k(t)]$ for integer values of $k$ between
$-5$ and $5$ and $-6 < t <4$ on the same graph.  Use the \cmd{view}
option of plot to only show what lies in the region $-10<y<10,
-10<z<10$, and use a sequence of colors so that each solution is a
different color. [ {\em HINT: you might find something like
\cmd{seq(COLOR(HUE,i/11),i=0..10)} useful for the latter.}]
}

\prob{16}{10/20}{
Find all the fixed points of the system
\begin{displaymath}
	\left\{
	\begin{array}{rcl}
		\dot{x} &\!=\!& x^2 + y, \\
		\dot{y} &\!=\!& x (y^2 - 1),
	\end{array}
	\right.
\end{displaymath}
a fixed point being a solution for which \emph{both $x(t)$ and $y(t)$}
stay constant. For each of these points, describe the behavior of the
solutions that have initial conditions nearby. You can use \m\ to
figure out what happens for nearby points, or you can use more
mathematical methods.}

\end{enumerate}

\bigskip
\par\noindent
\textbf{NOTE:} {\itshape The fact that there are various notations for
differential equations is purely intentional.}

\end{document}

