World Applied Sciences Journal 15 (12): 1774-1779, 2011 ISSN 1818-4952 © IDOSI Publications, 2011 Corresponding Author: Khaled Batiha, Prince Hussein Bin Abdullah College for Information Technology, Al al-Bayt University, Mafraq, Jordan 1774 A New Algorithm for Solving Linear Ordinary Differential Equations 1 Khaled Batiha and 2 Belal Batiha 1 Prince Hussein Bin Abdullah College for Information Technology, Al al-Bayt University, Mafraq, Jordan 2 Higher Colleges of Technology (HCT), Abu Dhabi Men's College, United Arab Emirates (UAE) Abstract: In this paper, a Differential Transformation Method (DTM) is used to find the numerical solution of the linear ordinary differential equations, homogeneous or inhomogeneous.The method is capable of reducing the size of calculations and handles linear equations, homogeneous or inhomogeneous, in a direct manner. Five examples are considered for the numerical illustrations of this method. The results demonstrate reliability and efficiency of this method for such problems. Key words: Differential transformation method • ordinary differential equations • Taylor's series expansion INTRODUCTION A variety of methods, exact, approximate and purely numerical are available for the solution of differential equations. Most of these methods are computationally intensive because they are trial-and- error in nature, or need complicated symbolic computations. The differential transformation technique is one of the numerical methods for ordinary differential equations. The concept of differential transformation was first proposed by Zhou [1] in 1986 [2-5] and it was applied to solve linear and non-linear initial value problems in electric circuit analysis. This method constructs a semi-analytical numerical technique that uses Taylor series for the solution of differential equations in the form of a polynomial. It is different from the high-order Taylor series method which requires symbolic computation of the necessary derivatives of the data functions. The Taylor series method is computationally time-consuming especially for high order equations. The differential transform is an iterative procedure for obtaining analytic Taylor series solutions of differential equations. The Differential transformation method is very effective and powerful for solving various kinds of differential equation. For example, it was applied to two point boundary value problems [6], to differential-algebraic equations [7], to the KdV and mKdV equations [8], to the Schrödinger equations [9] to fractional differential equations [10] and to the Riccati differential equation [11]. Jang et al. [12] introduced the application of the concept of the differential transformation of fixed grid size to approximate solutions of linear and non-linear initial value problems. Hassan [13] applied the differential transformation technique of fixed grid size to solve the higher-order initial value problems. The transformation method can be used to evaluate the approximating solution by the finite Taylor series and by an iteration procedure described by the transformed equations obtained from the original equation using the operations of differential transformation. The main advantage of this method is that it can be applied directly to nonlinear ODEs without requiring linearization, discretization or perturbation. Another important advantage is that this method is capable of greatly reducing the size of computational work while still accurately providing the series solution with fast convergence rate [14]. In this paper, we shall apply DTM to find the approximate analytical solution of the first, second and third order linear ordinary differential equation. Comparisons with the exact solution will be performed. THE DIFFERENTIAL TRANSFORMATION METHOD (DTM) An arbitrary function ƒ(x) can be expanded in Taylor series about a point x = 0 as: k k k k=0 x=0 x df f(x)= k! dx ∞       ∑ (1) The differential transformation of ƒ(x) is defined as: k k x=0 1 df F(x)= k! dx       (2) Then the inverse differential transform is