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{\Large\bf Introduction to First-Order Ordinary Differential Equations}

\vspace{.3in}

{\bf Applications}

\vspace{.2in}

Phenomena in many disciplines are modeled by first-order
ordinary differential equations (odes). Some examples include  

\begin{itemize}
\item Mechanical Systems
\item Electrical Circuits
\item Population Models
\item Newton's Law of Cooling
\item Compartmental Analysis 
\end{itemize}

\vspace{.3in}

{\bf General Form}

\vspace{.2in}

The general form of a first-order ordinary differential equation
is

$$
y'=f(t,y), \hspace{.2in} y(t_0)=y_0
$$

Here t is the independent variable and y(t) is the dependent variable.
The goal is to determine the unknown function y(t) whose derivative
satisfies the above condition and which passes through the point

$$
(t_0,y_0)
$$

\vspace{.3in}

{\bf Terminology}

\vspace{.2in}

\begin{itemize}
\item {\it ordinary differential equation}: An ordinary differential
equation (ode) is for which the unknown dependent variable is a
function of one variable only. For example: 

$$
y'=y^2+t
$$

is an ode. An example of a {\it partial differential equation} is

$$
u_{t}=u_{xx}
$$

Here the unknown dependent variable u(x,t) is a function of
both x and t. 

\item {\it order}: The order of an ode is the order of the
highest derivative in the equation. For example, the
ode

$$
y''=y'+\sin(y)
$$

is a second-order ode.

\item {\it linear}: A linear ode is one in which the dependent
variable and its derivatives appear linearly. Example of a linear
ode:

$$
y'=y\sin t + \log t
$$

This is a linear ode even though there are terms sin(t) and log(t).
The independent variable t can appear nonlinearly in a linear ode.
An ode that is not linear is nonlinear. Here is an example of a
nonlinear ode: 

$$
y'=t\sin y
$$

The general form of a linear ode is

$$
a(t)y'=b(t)y+c(t)
$$

\item {\it autonomous ode}: For an autonomous ode f(t,y) is a function
of y only. Here is an example of an autonomous ode:

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

\end{itemize}

\vspace{.3in}

{\bf Methods for Solving ODE}

\vspace{.2in}

There are three principal methods for analyzing and solving 
differential equations. These are 

\begin{itemize}
\item Qualitative analysis
\item Numerical integration
\item Analytic (exact) analysis
\end{itemize}

Most realistic odes {\it cannot} be solved exactly.
For these problems one does a qualitative analysis to get a
rough idea 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 with the answer obtained from qualitative
analysis. In a few fortunate cases a first-order ode can be
solved exactly. 

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