We know that when the coefficients and are analytic functions on a given domain, then the equation has analytic fundamental solution. We want to study equations with coefficients p and q having singularities, for this reason we study in this paper with one of the simplest cases, Euler’s equidimen- sional equation. This is an important problem because many differential equations in physical sciences have coefficients with singularities [3] . One of the special features of the equidimensional equation is that order of each derivative is equal to the power of the independent variable. This means that this type of equations can be reduced to linear equation with constant coefficient by using a change of the form.

Many numerical methods were developed for this type of equations, specifically on Euler’s equations such that Laplace transform method and Adomian method [4] . The method proposed in this paper was first established by Zhou to solve problems in electric circuits analysis. In this work, the differential transformation method is applied to solver the Euler equidimensional equations and to illustrate this method, several equations of this type are solved [5] [6] .

A Euler equidimensional equation is a differential equation of the form

where are constants and is an n-th derivative of the function

and is a continuous function.

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. In fact, for, we introduce, therefore. Then, the

first and second derivatives of are related by the chain rule,

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 and which give the general solution but depending on the type of roots it has, i.e.,

a) If, real or complex, then the general solution of the Equation (2) is given by

b) If, then the general solution of the Equation (2) is given by

Differential transformation method (DTM) of the function is defined as

In (7), we have that is the original function and is the transformed function. The inverse differential transformation is defined as

but in real applications, function is expressed by a finite series and Equation (8) can be written as

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 [4] [5] [6] .

Theorem 1 If, then.

Theorem 2 If, then with constant.

Theorem 3 If, then.

Theorem 4 If, then.

Theorem 5 If, then, where

Theorem 6 (Cárdenas, P). If, then

with.

To illustrate the ability of the Zhou’s method [2] [7] for the Euler equidimensional equation, the next problem is provided and the results reveal that this method is very effective.

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, then and therefore and. So, the new IVP is given by

The exact solution of the problem (12) is. Taking the differential transformation of this problem we obtain

or

where and. Therefore, the recurrence Equation (13) gives:

・ ,

・ ,

・ ,

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

but since, then we obtain (see Figure 1)

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 and therefore, so we obtain and. Then, the IVP is given by

The exact solution of the problem (15) is. Now, the DTM of (17) is

or

with and. So, the recurrence Equation (18) gives:

・ ,

・ ,

・ ,

・ ,

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

But since, then we obtain (see Figure 2)

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

Now, to find we use the chain rule. In fact we obtain

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

Now, as in the previous example and then. So, the new initial con- ditions are given by and. Using (7) we find that

and. Therefore, we obtain the IVP

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, then we obtain (see Figure 3)

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 [4] [5] and [6] clearly show the high efficiency of DTM with the three examples proposed.

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