\documentclass[12pt]{article}
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\begin{document}

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



\bigskip



\begin{enumerate} 
\item
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.}]

  
\item Consider the first order system
\begin{displaymath}
	\left\{
	\begin{array}{rcl}
x'&\!=\!&-2x+3y\\
y'&\!=\!&-3x
\end{array}
	\right.
\end{displaymath}
\begin{itemize}
\item{a} Find the fixed points.
\item{b} Determine the 'shape" of the solutions around the fixed point (that is, study whether it is a saddle, a source, a spiral source, a sink, a spiral sink or a center)
\item{c} Determine whether
\begin{displaymath}
	\left\{
	\begin{array}{rcl}
x(t)&\!=\!&e^{-t}\cos(2\sqrt{2}t)+2\sqrt{2}e^{-t}\sin(2\sqrt{2}t))\\
y(t)&\!=\!&3e^{-t}\cos(2\sqrt{2}t)
\end{array}
	\right.
\end{displaymath}
a solution of the system. Justify your answer.
\end{itemize}




\item (no \m)Consider a linear system $X'=AX$, where $A$ is a real 2x2 matrix. Show that if $V(t)=(v_1(t),v_2(t))$ and $U(t)=(u_1(t),u_2(t))$ are two solutions and $r$ and $s$ are two real numbers, then $W(t)=r V(t) + s U(t)$ is also a solution.

\item
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}


\end{document}

\item Consider the first order system
\begin{displaymath}
	\left\{
	\begin{array}{rcl}
x'&\!=\!&2x-3y\\
y'&\!=\!&-3y
\end{array}
	\right.
\end{displaymath}
\begin{alphlist}
\item Find two solutions of the system that are on a straight line.
\item Determine the 'shape" of the solutions around the fixed point (that is, study whether it is a saddle, a source, a spiral source, a sink, a spiral sink or a center)
\end{alphlist}