\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 of First-Order ODE}

\vspace{.3in}

Qualitative analysis is the first step in solving first-order
ode. Qualitative analysis can be used to verify numerical 
and analytic solutions. Even if an explicit formula is known,
qualitative analysis is useful, since it can give a visual
picture of the behavior of solutions to an ode.
We consider autonomous and non-autonomous ode separately.

Remember an autonomous first-order ode has the form
y'=f(y). The derivative is a function of the dependent
variable only. Here is an example:

$$
y'=4y(1-y)
$$

A non-autonomous first-order ode has the more general form
y=f(t,y). The derivative is a function of both the dependent
and independent variables. Here is an example:

$$
y'=t^2-y
$$

\end{document}
