In this paper, the generalized shifted Legendre polynomial approximation on an arbitrary interval was designed to find an approximate solution of a given second order nonlinear two-point boundary value problems of ordinary differential equations of the form

                          (1) 

Subject to the boundary conditions: 

                                                                            (2) .

The unknown Legendre coefficients of these nonlinear systems are the solutions of the system and they have been solved by continuation method. These unknown Legendre coefficients are then used to write the approximate solutions to the second order nonlinear two-point boundary value problems.

Legendre Polynomials are defined on the interval [−1,1] and can be determined with the aids of the following recurrence formulae [5]. 

      𝑟 = 1,2,3,                (3) 

In order to use this polynomials over arbitrary interval  with , we define the generalized shifted Legendre polynomial by introducing   Let the generalized shifted Legendre polynomial   be denoted by  Then   can also be obtained as : 

(𝑥) = 1 , (𝑥)    

(𝑥) =       𝑟 = 1,2,3, . . .   (4)  

The advantages of this method is that it needs less computational time and effort and it can be applied to find the solution of second order nonlinear two-point boundary value problems defined on any arbitrary interval. The results are illustrated by tables and graphs assisted by MATLAB.

Keywords: Boundary Value Problems (BVPs); Generalized Shifted Legendre Polynomials; Homotopy Continuation Method; Legendre Operational Matrix of Differentiation; Nonlinear Ordinary Differential Equations.