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solve exactly, and one of two approaches is taken to approximate the solution. The first approach is to simplify

the differential equation to one that can be solved exactly and then use the solution of the simplified equation to

approximate the solution to the original equation. The other approach, which we will examine in this paper, uses

methods for approximating the solution of original problem. This is the approach that is most commonly taken

since the approximation methods give more accurate results and realistic error information. Numerical methods

are generally used for solving mathematical problems that are formulated in science and engineering where it is

difficult or even impossible to obtain exact solutions. Only a limited number of differential equations can be

solved analytically. There are many analytical methods for finding the solution of ordinary differential equations.

Even then there exist a large number of ordinary differential equations whose solutions cannot be obtained in

closed form by using well-known analytical methods, where we have to use the numerical methods to get the

approximate solution of a differential equation under the prescribed initial condition or conditions. There are

many types of practical numerical methods for solving initial value problems for ordinary differential equations.

In this paper we present two standard numerical methods Euler and Runge Kutta for solving initial value prob-

lems of ordinary differential equations.

From the literature review we may realize that several works in numerical solutions of initial value problems

using Euler method and Runge Kutta method have been carried out. Many authors have attempted to solve ini-

tial value problems (IVP) to obtain high accuracy rapidly by using numerous methods, such as Euler method and

Runge Kutta method, and also some other methods. In [1] the author discussed accuracy analysis of numerical

solutions of initial value problems (IVP) for ordinary differential equations (ODE), and also in [2] the author

discussed accurate solutions of initial value problems for ordinary differential equations with fourth-order Runge

kutta method. [3] studied on some numerical methods for solving initial value problems in ordinary differential

equations. [4]-[16] also studied numerical solutions of initial value problems for ordinary differential equations

using various numerical methods. In this paper Euler method and Runge Kutta method are applied without any

discretization, transformation or restrictive assumptions for solving ordinary differential equations in initial val-

ue problems. The Euler method is traditionally the first numerical technique. It is very simple to understand and

geometrically easy to articulate but not very practical; the method has limited accuracy for more complicated

functions.

A more robust and intricate numerical technique is the Runge Kutta method. This method is the most widely

used one since it gives reliable starting values and is particularly suitable when the computation of higher de-

rivatives is complicated. The numerical results are very encouraging. Finally, two examples of different kinds of

ordinary differential equations are given to verify the proposed formulae. The results of each numerical example

indicate that the convergence and error analysis which are discussed illustrate the efficiency of the methods. The

use of Euler method to solve the differential equation numerically is less efficient since it requires h to be small

for obtaining reasonable accuracy. It is one of the oldest numerical methods used for solving an ordinary initial

value differential equation, where the solution will be obtained as a set of tabulated values of variables x and y.

It is a simple and single step but a crude numerical method of solving first-order ODE, particularly suitable for

quick programming because of their great simplicity, although their accuracy is not high. But in Runge Kutta

method, the derivatives of higher order are not required and they are designed to give greater accuracy with the

advantage of requiring only the functional values at some selected points on the sub-interval. Runge Kutta me-

thod is a more general and improvised method as compared to that of the Euler method. We observe that in the

Euler method excessively small step size converges to analytical solution. So, large number of computation is

needed. In contrast, Runge Kutta method gives better results and it converges faster to analytical solution and

has less iteration to get accuracy solution. This paper is organized as follows: Section 2: problem formulations;

Section 3: error analysis; Section 4: numerical examples; Section 5: discussion of results; and the last section:

the conclusion of the paper.

2.ProblemFormulation

In this section we consider two numerical methods for finding the approximate solutions of the initial value

problem (IVP) of the first-order ordinary differential equation has the form

0

00

,, ,

n

yfxyxxxx

yx y

(1)

where dd

yx

and

,

xyx is a given function and

x is the solution of the Equation (1). In this