On a first order linear singular differential equation in the space K’

Abstract

We propose in this work to describe all the generalized-function solutions of the non-homogeneous first-order linear singular differential equation with A, B two real numbers, s and p∈ N, n ≥ 1， q∈Z+, in the space of generalized functions K’. In the case of a second right-hand side consisting of an s-order derivative of the Dirac-delta function, we have completely investigated the considered equation when we look for the solution in the form of y(x)=∑k=0NCkδ(k)(x), with the unknown coefficients Ck which we have determined case by case, taking into account the relationship between the parameters inside. On the basis of what has been done, we focus our present research on applying the principle of superposition of the solutions that is conducting us to the awaited result when we also maintain the classical solutions of the homogeneous equation which remains the same.