TO SOLVING SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS WITH SINGULAR POINTS BY ADOMIAN DECOMPOSITION METHOD

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. The Adomin Decomposition Method (ADM) is used to solve differential equations, so in this study we used (ADM) to solve second order ordinary differential equations with singular initial value problem then, the equation was given a generalization.


INTRODUCTION
Second-order ordinary differential equations are one of the most widely studied classes of differential equations in mathematics, physical science, and engineering [5]. Ordinary differential equations with singular points may have solutions which are not analytic at those points, so series solution might not exist there [6]. This is because the solution may not be analytic at point and hence without having a series expansions about the point. In stead, we must use a more general series expansions. A differential equations may only have few singular points, but solution behavior near these singular points is important. The Adomian decomposition method has been paid much attention in the recent years in applied mathematics, and in the field of series solution particularly. Moreover, it is a fact that this method is powerful, effective, as well easily solves many types of linear or nonlinear ordinary or partial differential equations, and integral equations [8,2,3,4]. Many researchers are used this method to solve many kind of the differential equations such as Emden Flower Equation [7], first order ordinary differential equation [8]. We suppose the second ordinary differential equation with singular points as form: With initial conditions y(0) = A, y (0) = B, where A, B are constants and z(x, y) is known function.

DESCRIBE THE USER'S WAY
The equation (1) can be written as follow : And we have the inverse operator L −1 when we take L −1 for both sides of equation (2) we get with conditions The (ADM) is given the solution as series the Adomian polynomials A n are first constructed by Adomian, it gives general formula to determine the values of A n which gives the terms as: (6) and (7) in to (5), we get

Now in compensation
the y n (x) can by found as following : which gives Example (1): Consider the following equation y(0) = 0 and y (0) = 0.

Now the equation write it as
when we take L −1 to the last equation we get and the φ (x) = 0. Now the first value of y is the non-linear part is The Adomian polynomials for non-linear part e y are A 0 = e y o , Now we give the first terms from(9), (10) and (11) we obtain the solution in a series form (12) y(x) = y 0 + y 1 + y 2 = x 2 − x 4 60 + x 5 90 . x

GENERALIZATION
In this section, we will genralize equation (1) to the following form  Proof : We prove that by using mathematical induction : When m = 1 then the equation is written as then both sides give the same equations then the equation is hold.
Now we must prove the following formula Suppose that

DESCRIBE THE USER'S WAY
The equation (13) can be written as follow: (14) Ly = z(x, y), by using differential operator And we have the inverse operator L −1 when we take L −1 for both sides of equation (15) we get (17) y = φ (x) + L −1 z(x, y).
The equation can be written as When we take L −1 to both sides we get And φ (x) = e −x (1 + 2x + 2x 2 ). We obtain The Adomian polynomials for non-linear part Z(x, y) = 16e x x Ln(y) are Ln(y 0 ), which gives the first term (18) from (18), (19) and (20) we obtain the solution in a series form

CONCLUSION
In this work, we noticed the easy way for finding the approximate solutions to exact solutions, and we found its generalization and proved the generalization by using mathematical induction.
we discussed some examples to understand the method.