Solve Initial Value Problem Differential Equations

Solve Initial Value Problem Differential Equations-45
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.A difficult element in the analysis of this type of equations is the singularity behavior that occurs at x = 0.

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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.

The class of singular equations was generalized, by changing the coefficient of and the proposed technique was presented in a general way.

This gives the proposed scheme a wider applicability.

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).

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.

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.

The difficulty in using a taylor series method directly to this type of equations, due to the existence of singular point at x = 0, is overcome here.

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