On Solving Classes of Differential Equations with Applications

Given the difficultly of applying the methods of variation of parameters and undetermined coefficients for many classes of differential equations and inspiration of the role of the Linear Differential Operators to solve classes of Differential equations. In this paper, we introduce the nested factorization technique for solving classes of Differential equations using the basic differentiation and integration approach. Numerical examples with encouraging results have been presented to illustrate the efficiency of the method.


Introduction
Factorization of some operators in calculus such as the differential operators are strong computer algebra instrument for ordinary linear differential equations.This approach can be used for computing solutions and studying the structure of the differential (Van Hoeij, 1997).This paper is structured as follow.Section 2 discusses brief over-view and some fundamental rules of differential operator.In section 3, we present the nested factorization of the differential operator.Numerical example of well-known benchmark problem is presented in section 4. Finally, we present the conclusion and discussion in section 5.

Operator rules
In this research, the differential operators will be applied based on numerous rules they satisfy.We begin by assuming that the functions involved are sufficiently differentiable.This would make it easy to apply the operators.(i) Sum rule Suppose () and () are polynomial operators, then for any (sufficiently differentiable) function , Linearity rule For any constant   and functions  1 and  2 .Then, Multiplication rule If () = ()ℎ(), as polynomials in , then () = ()(ℎ()).
For more properties of the differential operator and their applications, please refer to (Mattuck, 2006;Van Hoeij, 1997).

The Nested Factorization of the Difference Operator
It has always been very difficult to find the general solution of the equation.
The convergence analysis of the procedure follows from the following Lemma.

Lemma 3
Let ,  ∈ ().The forward shift and the forward difference operators satisfy