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

Abstract

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<i> K’. </i>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 <i>A, B, C, m, n and r</i>. 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.