On Distributional Solutions of a Singular Differential Equation of 2-order in the Space K’

: The main purpose of this work is to describe all the zero-centered solutions of the second order linear singular differential equation with Dirac delta function (or it derivatives of some order) in the second right hand side in the space K’. All the coefficients and the exponents of the polynomials under the unknown function and it derivatives up to second order respectively, are real and natural numbers in the considered equation. We conduct investigations for both the euler case and left euler case situations of this equation, when it is fulfilled some particular conditions in the relationships between the parameters A, B, C, m, n and r . In each of these cases, we look for the zero-centered solutions and substitute the form of the particular solution into the equation. We then after, determinate the unknown coefficients and formulate the related theorems to describe all the solutions depending of the cases to be investigated.


Introduction
The importance of differential equations is well known as these equations describe many physical phenomena in our daily life. One also understand and know that it is not easy, in some specific cases, to solve certain kind of differential equations even those of the first order. Solving differential equations in the spaces of generalized functions such as K' and S ' is always challenging, and various scientific researches were devoted to such topic.
We recall that the theory of distributions was established by the imminent french mathematician Laurent Schwartz in 1945 posing the ideas which already in germ in the works of the great russian mathematician Sobolev in the 1930s. It is also well known that physics extended in space by functions of several variables and the expression of the laws of physics in terms of partial differential equations have been a great advance in the study of these phenomena. Talking to some specific notions, we know that sometimes even the concept of weak derivatives is not sufficient, and the need arises to define derivatives that are not functions, but are more general objects. Some measures and derivatives of measures will enter. We underline the fact that, for the purpose of setting up the rules for a general theory of differention where classical differentiability fails, Schwartz brought forward around 1950 the concept of distributions: a class of objects containing the locally integrable functions and allowing differentiations of any order.
In many scientific works it is mentioned that the distributional solutions, specifically as a series of Dirac delta function and its derivatives, have been used in several areas of applied mathematics such as the theory of partial differential equations, operational calculus, and functional analysis; in Physics such as quantum electrodynamics. We advice readers to see for more details the papers [15][16].
In our previous works, we have already used the Fourier Transform and its inverse applied to singular linear differential equation of some forms to describe and obtain all the generalized-function solutions of the considered equation, see [6,11,13]. In the same direction, we can quote among many others, for the generalized solutions, Nonlaopon et al. [17] used the Laplace transform technique to study some differential equations satisfying the differential equation.
Here we consider the following singular linear differential equation.
This is the first step we undertake to see in which way, we can generalize step by step, the results of researches obtained in the previous case studied for a linear singular differential equation of first order. See [6].
We structure this paper as follow: in section 2, we recall some fundamental well known concepts of distributions (generalized functions). Section 3 presenting the main results of the paper is firstly discribing, the Euler case and, secondly is devoted properly to the investigation of the solvability (existence of zero-centered solutions) of the considered homogenous equation in the situation called the left Euler case. We conclude our paper in section 4.

Preliminaries
Before we proceed to our main results, the following definitions and concepts well known from the theory of generalized functions are required. We also recall the notions of Fourier Transform and it inverse applied when looking for solutions of differential equations in our previous researches, for more details see [6].
By the way, we briefly review this important notions of Fourier transform, its properties, and generalized function centered at a given point (for a detailled study, we refer to [2,6,9,14]). We recall that K is denoted the space of test functions, of finite infinitely differentiable on R functions and K' the space of generalized functions on K.
For the function 1( ) ∈ 0, through 81 = 1 9 we noted the Fourier transform defined by the formula.
For the Fourier transform of generalized function, many properties are conserved as those taking place for Fourier transform for test functions, and particularly formulas of relationships between differentiability and decreasement.
Next, we need the following assertions which can be found with their proofs in the books for theory of generalized functions, for example see [2,3,9].
where & L are some constants.
Then it takes place the following formula.
As consequence from lemma 2.1 when N(") = " P , we obtain the following assertion.
The proof of this lemma can be found in some special mathematical books related to the theory of distributions, see also [2,3,9].
Sometime we used in our investigations the following very is called a generalized function (") ∈ 0′ that satisfies in the space 0′ the equality.
For the proof of this lemma it is sufficient to apply the definition 2.2, the Fourier transform to both members of the equation (8) and, next applying the inverse Fourier transform, we reach to the needed result.
Next, we consider a more complicated case when it is violated at least one of the two conditions (12).
First of all let consider the case = , + 2 ≠ , + 1 and, one of the situation to be investigated we call this case as following way. B. The left Euler case. In this section we consider the case = + 1, , > − 1 and call this situation of the equation (1) Next, let us calculate immediatly.
for that value of c G , and in the summation one term disappear. Therefore we have: Immediatly note that the right hand side in (30) -(31) are similar and therefore from these recurrent relationships, it is easy to obtain the general form of the coefficients | O . That allows us writing the result defined by formula (17).
The theorem is proved. The proof in the case c) is deduced from theorem 3.3, in the case b) from the fact that }( (y * ? ) (") = 0 , by the realization (35) and { * + − 1 − , < 0 and finally in the case a) it is obvious.

Conclusion
In this paper we have completly investigated the existence of the zero-centered solutions of the equation (1) in both cases called: Euler and left Euler cases.
We have look for the wanted solutions by replacing initialy the particular solution expressed with unknown coefficients into the initial equation (1) and, therefore we obtain in theorem 3.1 the necessary and sufficient conditions for the existence of zero-centered solutions of the equation in the euler case. Next, it is described all the solutions in the previous case within theorem 3.2 in connexion of the two possibilities mentionned. Investigating left euler case, we bring out the existence of non-trivial solutions in the form of Dirac delta functions as well as it derivatives up to the order − 2. The main results obtained and concerning this case, depending of the relationships between the parameters, are formulated in theorems 3.3 and 3.4.
From the obtained results, it is clear that it is challenging to try to imagine how to generalize such investigations of similar type of differential equation up to the general cases call left Euler case or right Euler case when there are realized the conditions: ∑