International Journal of Applied Science and Research 244 www.ijasr.org Copyright © 2021 IJASR All rights reserved Two Step Continuous Trigonometrically Fitted Method for Solving Oscillatory second Order Ordinary Differential Equations S.J.Kayode 1 , E.A.Areo 2 , J.O.Adegboro 3 1 Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria. 1,2,3 IJASR 2021 VOLUME 4 ISSUE4 JULY–AUGUST ISSN: 2581-7876 Abstract: The Continuous Two Step Trigonometrically-Fitted Second Order Method (TSTSOM) is used in this study to solve an oscillating problem of ordinary differential equations. The coefficients of the developed approaches are determined by the approximate solution’s frequency and step size, a discrete trigonometrically -fitted second order ordinary differential equation was recovered as a by-product. To demonstrate the method’s usefulness and efficiency, the method’s stability and other properties qualities will be described and implemented to solve linear and nonlinear initial value oscillatory problems. Keywords: Linearmultistep, interpolationtechniques, Trigonometric-fitted, predictor-corrector. 1. Introduction Mathematical modelling isacrucialtechnique for analyzing wide rangeofreal-world problems involving differential equations, spanning from physics, meteorology, and engineering to chemistry, biology, and social sciences. Differential equations are equations in which the dependent and independent variables have differential coefficients. Ordinary and partial differential equations are the two types of differential equations. Ordinary differential equations (odes) are differen-tiale equations in which the unknown parameter is afunction of one independent variable, whereas partial differential equations are those involving two or more independent variables (pdes).In Science and Engineering usually, mathematical models are developed to help in the understanding of physical phenomena. These models often yield equations that contains some derivative of an unknown function of one or several variables. In what follow, we consider a numerical solution of general second-order IVP soft he form  ′′ = (, ,  ′ ), ( 0 )= 0 , ′ ( 0 )=′ 0 ,  ∈ [ 0 ,  ] (1) where f satisfies the Lipschitz theorem. For the solution of a number of problems, numerical approaches based on the usage of polynomial functions have been presented (1).Adeniran and Ogundare(2015) offered a block hybrid technique for the direct integration of second order IVP whose solutions oscillate, and Ngwane and Jator (2013) proposed a hybrid block method for the system of first order IVP including oscillatory problems. Sanugi and Evans proposed the leap frog approach and the Runge-Kutta method, whereas Neta (1986) constructed families of backward differentiation equations. All of these techniques were implemented in a step-by-step manner. Despite their success, these methods have some drawbacks, including sensitivity to frequency changes, the necessity that the Jacobean’s Eigen values be wholly magi -nary, and computing burden. Psihoyios and Simos (2003 and 2005) proposed trigonometrically fitted schemes for the solution of oscillatory problems that are applied in predictor-corrector mode based on the well-known Adams-Bash forth method as predictor and Adams-Moulton as corrector, in the spirit of Kayode and Adegboro (2018) proposed predictor-corrector for solving second order ordinary differential equations.The method sarevery expensive to implement, require more human labor,andhavea lower level of accuracy. The purpose of this study is to develop a Discrete Trigonometrically Fitted Second Method (DTSM). This is accomplished by firstestablishinga TSCTM, which then gives adiscrete method that is used as a DTSMand uses the solution’s frequency as a priori knowledge. TSCTM, in partic ular, is made up of a collection of continuous functions, whereas DTSM is a by-product of TSCTM. Because the coefficients of the Continuous Trigonometric Second method TSCTM are functions of frequency and step size, the