Introduction

An ordinary differential equation (ODE) is an equation that involves one or more derivatives of an unknown function. A solution of a differential equation is a specific function that satisfies the equation.

In the following pages I will bring examples of a particular type of ODE - the initial-value problem for a first-order differential equation. The standard form is

x' = f (t, x) , x (a) is given.

It is understood that x is a function of t, so that that the differential equation written in more detail looks like this:

dx(t)/dt = f (t, x(t))

Although many methods exist for obtaining analytical solutions of differential equations, they are primarily limited to special differential equations. In practical problems, however, frequently a differential equation is not amenable to analytical solution and a numerical solution must be sought. Even when a formal solution can be obtained, a numerical solution may be preferable , especially if the formal solution is very complicated.