Many problems in engineering and science can be formulated in terms of differential equations.

An equation involving a relation between an unknown function and one or more of its derivatives is called a differential equation. These differential equations basically fall into two classes, ordinary and partial, depending upon the number of independent variables present in the differential equation; one for ordinary and more than one for partial differential equation.

We shall confine our attention here only to the numerical solution of ordinary differential equations (ODE's).

The general form of an ODE can be written as

(1.1)

Where is a differential operator and is a given function of the independent variable . The order of the differential equation is the order of its highest derivative. If the dependent variable and its derivatives occur in the first degree and are not multiplying each other, the equation is said to be linear, otherwise it is non linear. The general order non linear differential equation can be written as

(1.2)

This may be written in the form

(1.3)