Differential equations are considered as the main applicable solution for different problems not only in mathematics but also in several science branches. Differential equations are used in many different fields; because of that, these equations become very important for implementing several numerical approximations. The Runge-Kutta Method is preferred to be used when it is desired to solve analytically complex models by applying basic mathematical techniques. Therefore, this work aims to present several examples, mainly numerical, showing how approximate solutions of certain differential equations with initial conditions can be obtained using the methodology of the random Runge-Kutta and Euler methods. The study also introduced a numerical approximation of differential equations with randomness, convergence, and its applications. The results show that the numerical values of the Variance (Var(Xn)) and Mean Square (E(Xn)) are closer to the exact or theoretical values of (Var(Xn)) and (E(Xn)). Also, the results show that the approximations that are obtained by the method of Runge Kutta in order four give good results compared to those established by the Euler method when the decrease occurs in the step parameter (h).