On Classical and Distributional Solutions of a Higher Order Singular Linear Differential Equation in the Space K’

Abstract

In this research work, we aim to find and describe all the classical solutions of the homogeneous linear singular differential equation of order l in the space of K' distributions. Recall that in our previous research, the results of which have been published in some journals, we had undertaken similar studies in the case of a singular differential equation of the Euler type of second order, when the conditions were carried out. That said, our intentions in this article are therefore to generalize the results obtained and recently published, focusing our research on the situation of the homogeneous singular linear differential equation of order l of Euler type. In this orientation, we base ourselves on the classical theory of ordinary linear differential equations and look for the particular solution to the equation considered in the form of the distribution with a parameter to be determined, which we replace in the latter. Depending on the nature of the roots of the characteristic polynomial of the homogeneous equation we identify, case by case, all the solutions indicated in the sense of distributions in the space K'. In this same work, we return to the non-homogeneous equation of order l of the same Euler type, whose second member consists only of the derivative of order s of the Dirac-delta distribution studied in our previous work, to fully describe all the solutions of the latter in the sense of distributions in the space K'. We finalize this work by making an important remark emphasizing the interest in undertaking research of the same objective of finding a general solution, by studying the singular differential equations of the same higher-order l with the particularity of being of Euler types on the left and Euler on the right in the space of distributions K’.