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1. Introduction

Differential equations play an important role in many problem of engineering and science. This is so because of the involvement of differential equation in the mathematical models. It naturally leads to the determination of solution numerically. The use of the numerical methods have become vital in the absence of explicit solutions. Normally numerical methods have two major roles:
(a) the method should be implementable on a computer;
(b) the method should deal with the analysis of error estimates.
For the present we deal with some of the numerical methods to solve initial value problems (IVP's) (on finite intervals). We also mention that no effort is made to study boundary value problems. Let us consider an initial value problem.

where f, a and $y_0$ are prescribed and . At the moment we assume that the initial value problem (1.1) has a unique solution existing on [a,b] and we denote it by y. Our aim is determine y on the interval [a,b]. Let us also assume that our aim is to determine y at b. We divide the interval [a,b] into N equal parts.

$$a,a+h,...+ a + Nh = b$$

where $h=(b-a)/N$. h is called the step size and h depends on, a,b and N. The value of y at a+ih, $1\leq 1 \leq N$, is denoted by (ie $y_i = y...a...h$) and the approximation of y at $a+ih$ is denoted by . In this chapter we develop methods for determining . Since there may not be methods for finding $y(b)$, we shift emphasis to determine "an approximate" value (namely for ). This is achieved by evaluating the approximate values, $y_1,y_2,..y_{n-1}$. Normally is called the initial value, $x = a + h$ is called the first step, is called the $j^{th}$ step. $y_j$ is called the approximation of at the $j^{th}$ step. A method which uses to determine $y_j$(i.e the approximate value at the previous step, is called a single step method). Otherwise the method is called a multi-step method.
In the sequel we deal with some simple methods to find the approximate value of the solution of (1.1). In these methods our stress is on the use of computers for numerical evaluation. In other words, implementation of the methods by using a computer is one of our present aims.

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