% Dee Winn
%
% This is a LaTeX document that gives a brief introduction to
% second order ODE's.
%

\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 Introduction to Second-Order Ordinary Differential Equations}

\vspace{.3in}

{\bf Applications}

Many physical phenomena are modeled by second order ODE's.  Some
examples are:
\begin{itemize}
\item Mechanical Systems/Vibrations (springs, pendulums, etc)
\item Electrical Circuits
\item One Dimensional Motion
\item Pursuit Problems
\end{itemize}

\vspace{.3in}

{\bf General Form}

\vspace{.2in}

The general form for a second order ordinary differential equation is
$$
y''=f(x,y,y')
$$
on some interval [a,b]. Here, x is the independent variable and y(x)
is the dependent variable. The goal is to find functions that satisfy
the above ordinary differential equation. 

Usually, the ode is accompanied by initial conditions or boundary
conditions. Initial conditions have the form:

$$
y(t_0)=y_0 \hspace{.2in} y'(t_0)=y'_0
$$

Here t_0 is some starting time and y_0 and y'_0 are values of the
the function and its derivative at t_0. Alternatively, boundary
conditions can be specified. These involve specifying the function
and or its derivative at two different values of t. For example,

$$
y(t_0)=a \hspace{.2in} y(t_1)=b
$$

\vspace{.3in}

{\bf Methods for Solving Second-Order ODE}

\vspace{.2in}

There are three principal methods for analyzing and solving 
second-order differential equations. These are
\begin{itemize}
 \item Qualitative analysis
 \item Numerical analysis
 \item Analytic (exact) analysis
\end{itemize}
Most second-order odes arising in realist applications cannot
be solved exactly. For these problems one does a qualitative
analysis to get a rough ide of the behavior of the solution.
Then a numerical method is employed to get an accurate solution. 
In this way, one can verify the answer obtained from the numerical
method by comparing it the answer obtained from qualitative
analysis. In a few fortunate cases a second-order ode can be
solved exactly.

\end{document}
