Some remarks concerning the m-reduction of differential equations

Abstract

In Skórnik, Wloka [8] we gave an answer to the question, when it is impossiable to solve a linear differential equation of order n in terms of linear differential equations of lower order ≤ n-1. Because of isomorphy, see [8], the answer is also valid after eventually applying Laplace transformation or operational calculus. Here in this paper, by using the results of M.F. Singer [10], [11], we give a very simple derivation of our sufficient condition dim G > (n - 1)2 G the Galois group of the equation. It turns out, that for some little n′s, for instance n = 3,5, this condition is also necessary.