^{1}

^{2}

^{2}

In this work, we apply the Zhou’s method [1] or differential transformation method (DTM) for solving the Euler equidimensional equation. The Zhou’s method may be considered as alternative and efficient for finding the approximate solutions of initial values problems. We prove superiority of this method by applying them on the some Euler type equation, in this case of order 2 and 3 [2]. The power series solution of the reduced equation transforms into an approximate implicit solution of the original equations. The results agreed with the exact solution obtained via transformation to a constant coefficient equation.

We know that when the coefficients

Many numerical methods were developed for this type of equations, specifically on Euler’s equations such that Laplace transform method and Adomian method [

A Euler equidimensional equation is a differential equation of the form

where

and

Now, we consider a second order differential equation (homogeneous Euler equidi- mensional) of the form

The solution can be obtained by using the change of variables

where

first and second derivatives of

Now, substituting (4) in (2) yields a second order differential equation with constant coefficients, i.e.,

Equation (5) can be solved using the characteristic polynomial

where roots are

a) If

b) If

Differential transformation method (DTM) of the function

In (7), we have that

but in real applications, function

which implies that

is negligibly small where n is decided by the convergence of natural frequency in this study.

The following theorems that can be deduced from Equations (7) and (9) and the proofs are available in [

Theorem 1 If

Theorem 2 If

Theorem 3 If

Theorem 4 If

Theorem 5 If

Theorem 6 (Cárdenas, P). If

with

To illustrate the ability of the Zhou’s method [

Example 1 (Homogeneous case). To begin, we consider the initial value problem

Using the substitution (3) and (4), the IVP (10) is transformed to a second order differential equation with constant coefficients, i.e.,

Now, of the initial conditions we have that as

The exact solution of the problem (12) is

or

where

・

・

・

Therefore, using (9), the closed form of the solution can be easily written as

but since

Example 2 (Non-homogeneous case). We consider the following IVP

Then, problem (15) is transformed to a second order differential equation with con- stant coefficient by using (3) and (4), i.e.,

We know that of the initial conditions

The exact solution of the problem (15) is

or

with

・

・

・

・

Therefore, using (9), the closed form of the solution can be easily written as

But since

Example 3 (Third order Euler’s equation). Consider the following IVP

Now, to find

Therefore, using (3), (4) and (21) we have

Now, as in the previous example

Applying DTM to (23) we obtain

or

So, the recurrence equation (24) gives:

・

・

・

Therefore, using (9), the closed form of the solution can be easily written as

But since

In this paper, we presented the definition and handling of one-dimensional differential transformation method or Zhou’s method. Using the substitutions (3) and (4), Euler’s equidimensional equations were transformed to a second and third order differential equations with constant coefficients, next using DTM these equations were transformed into algebraic equations (iterative equations). The new scheme obtained by using the Zhou’s method yields an analytical solution in the form of a rapidly convergent series. This method makes the solution procedure much more attractive. The figures [

Foremost, we would like to express my sincere gratitude to the Department of Mathematics of the Universidad Tecnológica de Pereira and group GEDNOL for the support in this work. In the same way, we would like to express sincere thanks to the anonymous reviewers for their positive and constructive comments towards the improvement of the article.

Cárdenas Alzate, P.P., Salazar, J.J.L. and Varela, C.A.R. (2016) The Zhou’s Method for Solving the Euler Equidimensional Equation. Applied Mathematics, 7, 2165-2173. http://dx.doi.org/10.4236/am.2016.717172