A Semi-Analytical Method for Solutions of a Certain Class of Second Order Ordinary Differential Equations

This paper presents the theory and applications of a new computational technique referred to as Differential Transform Method (DTM) for solving second order linear ordinary differential equations, for both homogeneous and nonhomogeneous cases. For the robustness and efficiency of the method, four examples are considered. The results indicate that the DTM is reliable and accurate when compared to the exact solutions of the solved problems.


Introduction
Most of the problems encountered in applied sciences, management and economics take the forms of second order linear ordinary differential equations. Sometimes, obtaining exact solutions through the direct methods (analytical methods) for these systems seems difficult even if the exact solutions exist, hence the need for numerical techniques for approximate solutions. Some of these numerical methods involve linearization, disscretization and perturbation, and they only permit the solutions to a given ODE at a certain interval. In addition, they are intensive in terms of computation and as such, lead to the situations where some basic phenomena are technically avoided.
The notion of DTM was first introduced by Zhou [1] while solving linear and nonlinear initial value problems in electric circuit analysis. The Method provides an analytical approximate solution to linear and nonlinear

The Differential Transform Method
This section introduces the basic concepts and theorems of DTM needed for applications in the remaining sections. = , then the one-dimensional th k differential transform of the ( ) f x defined as ( ) F k is given as: Equation (1) is the transformed function of ( ) f x . Definition 2. The differential inverse transform of ( ) F k is a Taylor series expansion of the function ( ) , defined as : Combining (1) and (2) yields:

Some Basic Theorems of the Differential Transform Method
The following theorems and properties of the DTM are stated below for the sake of applications, their proofs can be found in [14] and [15]. Let respectively, with, α ∈  and δ a Kronecker delta, then the following theorems hold: In particular, we have:

The DTM and the Second order Linear Ordinary Differential Equations (ODEs)
In this section, we present clearly how a second order linear ODE with constant coefficients is transformed using the DTM. The corresponding ODE is of the form: Thus, (4) becomes We will take the differential transform (DT) of (6) by applying theorems (2.1-2.6) as follows: subject to the initial conditions ( ) ( ) ( ) (9) is a recursive formula for the computation of coefficient terms in the series solution of the problem. Therefore, using (2) and (9) gives the approximate solution of (4) as:

Applications and Numerical Results
In this section, we will apply the discussed DTM to solve some problems whose results will be compared with the theoretical (exact) solutions.
subject to ( )

Procedure
We rewrite (11) in a standard form and take the differential transform (DT) as follows: with the initial conditions ( ) ( ) By using the recursive relation in (15) with 0 k ≥ , we obtain values for ( ) ( ) ( ) Equation (16) is the same with the theoretical solution in (12).

Procedure
We re-write (16) in a standard form and take the differential transform as follows: with the initial conditions ( ) ( ) using (19) gives the following: We observed that ( ) ( ) ( ) ( ) . Hence, the solution of (17) by (10) is reformed as: Equation (20) As such, from (10), the solution of (21) is: Procedure Equation (27) by the differential transform method becomes; subject to the initials ( ) ( ) Thus, for 0 k ≥ , the values of ( ) ( ) ( ) 2 , 3 , 4 , Y Y Y  , obtained using (29) are given below: For Hence, from (10), the solution of (27) is:  We present numerical comparisons between the exact, and the numerical solutions based on a 5-iterate DTM using the coefficient terms from Table 1, Table 2, Table 3, and Table 4 as shown in Table 5, Table 6, Table 7, and Table 8 respectively.

Discussion of Results and Conclusion
In this paper, we have presented a semi-analytical method (DTM) for solving a certain class of ODEs. The DTM has advantages over other numerical techniques as it does not involve linearization, discretization or perturbation of a given problem; hence it has no effect of computational round off error. The DTM also provides a closed-form solution; therefore, it is very powerful and effective in finding both analytical and numerical solutions of second order linear ODEs with constant coefficients.  (15).