Abstract: Differential equations are commonly used for mathematical modelling in science and engineering. In most real life situations, the differential equation that models the problem is complex to solve exactly, only a limited number of differential equations can be solved analytically. Large number of ordinary differential equations whose solutions cannot be obtained by regular analytical methods where we have to use the numerical methods to get the approximate solution of a differential equation under the prescribed initial conditions. This paper mainly presents Eulerís method and fourth-order Runge -Kutta Method (RK4) for solving initial value problems (IVP) for ordinary differential equations (ODE). In order to verify the accuracy, compare numerical solutions with the exact solutions. This paper will then proceed to explain what steps the methods actually carries out in solving the differential equation along with the high level programming language code. The programme result is in good agreement with the numerical solutions and exact solutions.

Keywords: Initial value problem, Eulerís Method, Runge - Kutta Method.