The Runge-kutta scheme which is a one-step method remains one of the most versatile Numerical schemes used in the solution of Initial value problem(IVP) in the real world of Engineering, Biological and Social sciences more especially some stiff problems. In this Paper we present some One-step schemes such as Euler’s (RK₁) Heun’s Method (RK₂) Runge-kutta of IV (RK₄) and Runge-kutta of V (RK₅) for the unique solutions of some problems in ordinary differential equation. The performance of these methods was critically analyzed at various stages of mesh points (h) with respect to difference methods. The performance of h of different method provide a convincing superiority, uniqueness, efficiency as well as the theoretical stability of each method to the solution of Initial value problems in ode. This was carefully achieved by the schemes or methods satisfying the uniqueness, consistence and stability condition. A computational numerical comparism on each table shows that at given an adjusted step size h, to the advantage backward Euler’s method (h/4) Heun’s method with a step size (h/2), RK₄ and RK₅ methods still perform and produce better and stable results as they converge to the exact solution.

IVP, Real Life Problems, One-Step Methods (Runge-Kutta of Orders I II, IV and V)