Solve Initial Value Problem Differential Equations

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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.

Moreover, three-point difference method of second order have been also used by Chawal and Katt I(1984), Chawla et al. However, Jain and Jain (1989) derived three-point difference method of four and six order to solve this problem.

The numerical results obtained in Jain and Jain (1989) demonstrate o (h) convergence of the method. Introduction to Nonlinear Differential and Integral Equations.

Russell and Shampine (1975) have investigated (2) for the liner function f (x, y) = ky h (x) and have proved that a unique solution exists if h (x) ∈ C [0,1] and –∞4.

Three-point difference methods of second order have been used in Russell and Shampine (1975). Multi-integral method for nonlinear boundary-value problems: A fourth order method for a singular two point boundary value problem. 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).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).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. Solving Singular Initial Value Problems in the Second-order Ordinary Differential Equations. DOI: 10.3923/jas.2007.2505.2508 URL: doi=jas.2007.2505.2508 INTRODUCTION In recent years, the studies of singular initial value problems in the second-order Ordinary Differential Equations (ODEs) have attracted the attention of many mathematicians and physicists.Equation 2 differs from the classical Lane-Emden-type Eq.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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