% Dee Winn
% 
% This is a LaTeX file describing second order linear ODEs qualitative 
% analysis.
%

\documentstyle[11pt]{article}

\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\topmargin}{-.2in}
\setlength{\textheight}{9.5in}
\setlength{\headheight}{0in}
\setlength{\headsep}{0in}
\setlength{\parindent}{0pt}
\setlength{\parskip}{5pt}

\begin{document}
\pagestyle{empty}

{\Large\bf Qualitative Analysis for Second Order ODE's}

\vspace{.3in}

Qualitiative analysis aids in the understanding of general behavior
of solutions without necessarily solving the given ODE.  Properties
such as the solution's long time behavior, periodicity, and stability
toward steady states can be discerned with proper qualitative analysis.

As mentioned, most ODE's, especially nonlinear ODE's, cannot be solved
exactly.  For a second order ODE,
$$ y''(x) = F(t,y,y') $$
one can write the equation as a system of two first order equations by
making the substitution v=y'.  Then compute approximate solutions
for various initial conditions 
\mbox{y(0) = a, v(0) (=y'(0)) = b}
using first order numerical methods.  The solution to the system is a
vector \mbox{( y, v ) = ( y(t), v(t) ).}  This vector can be plotted
parametrically in the {\em phaseplane} or yv-plane.

{\bf Example :}
Consider the nonlinear simple pendulum equation
$$ y'' = -\omega^{2}sin(y). $$
Substituting v = y', gives
$$ y' = v $$
$$ v' = - \omega^{2}sin(y). $$
Using Matlab to compute the approximate solutions and the plots yields:
(remember - the solution is a vector (y,v) where y is the position
and v is the velocity.  The arrows in the plot are the various vectors
(y,v) plotted parametically.)

\vspace{.3in}







\vspace{.3in}

Can you trace out one trajectory (the path of a single solution) ?
Closed trajectories imply periodic solutions.  The solution with
y(0) = 1, v(0) = 0 is plotted here as a closed curve:

\vspace{.3in}

\vspace{.3in}

%If one adds damping to the equation
%$$ y'' = -cy' - \omega^{2}sin(y), \: \: c>0$$
%then one obtains the following phaseplane picture.  Can you explain
%the different trajectoies seen in the picture? A few trajectories
%are also plotted.
% 
%\vspace{.3in}
%
%\vspace{.3in}

\end{document}
