Abstract
The study presents a new method for solving initial value problems (IVPs) for ordinary differential equations (ODEs). The study successfully satisfied Butcher conditions and got a Jacobian matrix: a(11), a(12), a(13), a(21), lambda, a(23), a(31), a(32), lambda and then applying modified Newton, to find out the unknown vector lambda(i) from iterative equation y(i) = y(i) + partial derivative(i).The diagonal-implicit Range-Kutta, method submitted by this study solves the problem arising from application of an implicit multistage for linear or nonlinear algebra problems.The results achieved by using these parameters via Visual studio C++, compared to the ordinary differential equations that have an exact solution are more than impressive and get a shouted result, for example while comparing the exact solution of an ordinary differential equation with our numerical solution for the same problem, we get an approximation error 9.9873D-05. Therefore, with great confidence we can use this method with problems that need numerical solution.