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Abstract: This paper is involved with the take a look at at the Runge-Kutta method to use on a special order of differential equation and solve special varieties of hassle consisting of initial price trouble and boundary fee trouble inside the ordinary differential equation. at first, we discuss the definition and era of the differential equation in particular based at the partial differential equations after which the definition of the Runge-Kutta method and the derivation of the midpoint approach, and the formula of the Runge Kutta method of fourth order and sixth order. We also write FORTRAN ninety/95 software for one-of-a-kind order of Runge-Kutta methods. we have solved a few examples of the fourth-order RK technique and sixth-order R-k method to get the application of R-ok approach. We also compared the solution of R-k method with the precise answer for distinctive step sizes. Then we have given simultaneous first-order differential equation and 2nd-order differential equation and then solved them by way of fourth order Runge-Kutta approach. At remaining, we've discussed the boundary price problem which we have solved by way of the fourth and sixth order R-k method. After that, we've got written the algorithm of the shooting technique and confirmed laptop consequences with the distinction among two solutions at the side of probabilities of mistakes.