Many mathematical problems arising from physical phenomena are nonlinear in nature, particularly problems expressed in the form of differential equations. It is known that only a very few higher order nonlinear ordinary differential equations have exact solutions. Thus the need arises to study the equations numerically and almost all of these nonlinear problems would require some sort of numerical solution. In 2003, a completely new way of finding numerical solutions for higher order nonlinear non-stiff differential equations involving initial values was introduced by Zhu and Phan. This method was then modified by Zhu and Hammel in 2006 to improve the accuracy of the original method. In this project report we look at the accuracy of these methods for different higher order nonlinear non-stiff ordinary differential equations with initial conditions, test problems. The performances are then compared with the popular class of methods such as Runge-Kutta methods, multi-step methods and predictor-corrector methods.