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\begin{document}

\begin{center}
{\large Math 331, Fall 2004: Problems Set 6, Due Oct 25}
\end{center}

\bigskip
%


\begin{enumerate} 
\item 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.}]


\item
For the functions $y_k(t)$ and $z_k(t)$ found in the previous problem 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.}]



\item {
Consider the differential equation
$\dot{\mb{z}}(t) = \mb{F}(\mb{z}(t))$, where the vector $\mb{z}(t)
=(x(t),y(t))$ and the field $\mb{F}(x,y) = (-y,x-y)$. Plot a few
solutions. What happens to them when $t \to +\infty$?
Give a ``\m-proof'' that this is a general fact for \emph{every}
solution. [{\em A ``\m-proof'' is an argument that is rigorous once we
accept \m\ results as incontrovertibly true.}]
}


\item {
For the system of differential equations of
problem set 5,
\begin{displaymath}
        \left\{
        \begin{array}{rcl}
                \dot{x} &\!=\!& x^2 + y, \\
                \dot{y} &\!=\!& x (y^2 - 1),
        \end{array}
        \right.   
\end{displaymath}
find the eigenvalues and eigenvectors of the Jacobian at the fixed points.

}


\item {
 Consider the equations of the glider with no drag
term ($R=0$). Use \cmd{dsolve, type=numeric} to solve them numerically
with initial conditions $\theta(0)=0$, $v(0)=0.8$. Then solve exactly the 
linearized system around the fixed point $(\theta_0,v_0) = (0,1)$, with
the same initial conditions. Graph the two functions for $0 \le t \le 5$,
and give a good estimate of their maximum difference. What happens if we
take a larger $t$-range?
}

\end{enumerate}
\end{document}