Australian Journal of Basic and Applied Sciences, 5(12): 2121-2126, 2011 ISSN 1991-8178 Corresponding Author: Kayode, S.J., Department of Mathematical Sciences, Federal University of Technology, P.M.B. 704, Akure, Ondo State, Nigeria. E-mail: cosjkay57@yahoo.com, GSM: +2348033974438. 2121 A 3-Step Hybrid Method For Direct Solution Of Second Order Initial Value Problems Kayode, S.J. and Adeyeye, O Department of Mathematical Sciences, Federal University of Technology, P.M.B. 704, Akure, Ondo State, Nigeria. Abstract: This paper considers the development of a 3-step two-point continuous hybrid method for solving general second order initial value problems (IVPs) of ordinary differential equations using Chebyshev polynomial series as basis function. The method is implemented by using main predictor of the same order which was derived through the same procedure. Both the method and its predictor are consistent and zero-stable. Numerical results show a superior accuracy of the method when compared with the existing schemes of the same order. Key words: second order initial value problems; two-point continuous hybrid; Chebyshev polynomial series; predictor-corrector mode. INTRODUCTION In this paper, a direct numerical solution to the general second order initial value differential equations of the form ' 0 0 0 0 '' (, , '), ( ) , '( ) y f x yy y x y y x y    (1) is proposed (where f is a continuous and differentiable function) without adopting the conventional way of reducing it to a system of first order equations and then applying the various methods available for solving systems of first order IVPs. This approach of reducing (1) to systems of first order equations however has many disadvantages as discussed in different literature such as (P. Onumanyi, 1994; S.N. Jator, 2001; D.O. Awoyemi and S.J. Kayode, 2002; Z.A. Majid, 2009). These disadvantages include computer programs associated with the methods are often complicated especially when incorporating subroutines to supply the starting values for the methods resulting in longer computer time and more computational work. Considerable efforts have been devoted to the development of various methods for solving (1) directly by various authors such as (D.O. Awoyemi, 2001; D.O. Awoyemi and S.J. Kayode, 2005; A.O. Adesanya, 2008; Z.A. Majid, 2009) and (S.J. Kayode, 2010; Y.A. Yahaya and A.M. Badmus, 2009) developed a class of hybrid methods for solving (1) but the methods are of low order of accuracy. (S.J. Kayode, 2011) developed a class of one-point hybrid implicit methods whose order of accuracy is higher is higher than (Y.A. Yahaya and A.M. Badmus, 2009). In this paper, a two-point continuous hybrid method of higher order of accuracy is developed for direct approximation of the solution of (1). The Derivation Of The Method: In this section, we apply the interpolation and collocation procedures and we choose our interpolation points (i) at grid points and our collocation points (c) at both grid and two off-grid points. We consider a partial sum of Chebyshev series in the form ( ) 0 () () c i n n n yx aT x     (2) as an approximate solution for the development of the method, where j a 's are the parameters to be determined and () n T x is the Chebyshev polynomial of first kind. Obtaining the second derivative of (2), gives '' ( ) 0 () () n c i n n y x aT x      (3)