# Solve Initial Value Problem Differential Equations

A difficult element in the analysis of this type of equations is the singularity behavior that occurs at x = 0. THE METHOD In theory, the infinite Taylor series can be used to evaluate a function, given its derivative function and its value at some point, consider the nonlinear first-order ODE: . To illustrate the generalization discussed above, we discuss this example: Example 3. CONCLUSION In the discussion it was shown that, with the proper use of the taylor series method, it is possible to obtain an analytic solution to a class of singular initial value problems, homogeneous or inhomogeneous.

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We are interested in the existence of solutions to initial-value problems for second-order nonlinear singular differential equations.

We show that the existence of a solution can be explained in terms of a more simple initial-value problem.

1 for the function f (x, y) and the inhomogeneous term g (x).

Equation 1 with specializing f (y) was used to model several phenomena in mathematical Physics and astrophysics such as the theory of stellar structure, the thermal behavior of a spherical cloud of gas, isothermal gas spheres and theory of thermionic currents Chandrasekhar (1976) and Davis (1962).

Mathematics and Computing Department, Beykent University, Ayazağa, Şişli, 34396 Istanbul, Turkey Received 4 December 2013; Accepted 10 March 2014; Published 9 April 2014Academic Editor: Elena Braverman Copyright © 2014 Afgan Aslanov.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Moreover, a generalization was developed in Wazwaz (2001) by replacing the coefficient 2/x of (x) by n/x. It is important to note that (2), with boundary conditions, has attracted many mathematicians and has been studied from various points of view. We consider the new auxiliary (nonhomogeneous, but easily solvable) (4) instead of (42).The conditions we obtained are weaker than the previously known ones and can be easily reduced to several special cases.Due to the significant applications of Lane-Emden-type equations in the scientific community, various forms of f (y) have been investigated in many research works.A discussion of the formulation of these models and the physical structure of the solutions can be found in Chandrasekhar (1976), Davis (1962), Shawagfeh (1993), Adomian (1986) and Wazwaz (2001).Local existence and uniqueness of solutions are proven under conditions which are considerably weaker than previously known conditions.The problem of the existence of a solution is reduced to the finding of a solution of some more easy problems like (4).It is convenient to introduce the derivative of order k. 5, we can get the higher derivatives for y in the following Hence x = 0, y (0) = A, (0) = B, we can find By taylor series method with x = 0. We have substituting the initial condition and the values find, we obtain the solution.

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