Given a differential equation with the condition ), where is any point in the domain of y, we can define an initial value problem as a problem whose solution is being sought for at a particular point. Therefore an initial value problem is of the form; , ), for where is the domain of .In this work we shall concern ourselves with the analysis of first order ordinary equations of initial value type which are of the form = with initial condition ) for using Runge Kutta methods which are one step methods.Our focus will be on a class of methods under the one step method called the RungeKutta method which is an important family of implicit and explicit iterative methods for the approximation of solutions of initial value problems of ordinary differential equations.Numerical methods for solving initial value problems of first order ordinary equations in recent times have been applied to stiff problems which is an important class of ODE and are generally produced from problems with very different time scales (usually when one component decays more rapidly than the other). In a subsequent chapter we shall see how the various RungeKutta method discussed in this work behave whenever they are applied to a stiff problem. KEYWORDS RungeKutta Methods, Initial Value Problems, Explicit and Implicit Methods, Consistency, Stability, Convergence

Keywords: RungeKutta Methods, Initial Value Problems, Explicit and Implicit Methods, Consistency, Stability, Convergence