A special type of two step Runge-Kutta method for solving first and second order ordinary differential equations(ODEs)are proposed.The method arise from the classi cal Runge-Kutta method for approximating the numerical solution of first order initial value problems and is developed for solving special second order ODEs which is noted as improved Runge-Kutta Nystrom method.With the aim to increase the computational efficiency,the methods are obtained of higher order with less number of stages and function evaluations.The convergence and stability properties of methods are discussed.The numerical results of the methods based on error accuracy and number of function evaluation are compared with the existing classical type of Runge-Kutta methods for solving first and second order ODEs which indicate that the special type of two-step Runge-Kutta methods computationallv are more efficient.