Problems that appear during factorization of ordinary linear differential operators (Q1894544)

Let \(\mathbb{K}\) be a differential field of characteristic 0, \(L=D^n+f_1(x) D^{n-1}+ \dots+ f_n(x)\) be a linear ordinary differential equation where \(D=d/dx\), \(f_s(s) \in \mathbb{K}\). The problem of representing \(L\) in the form of the composition \(L=L_k\cdot \dots \cdot L_1\) of operators of the same form, but of lower orders with coefficients \(f_{is} (x)\in\mathbb{K}\) is considered. The method proposed by Beke and Schwarz [see \textit{E. Beke}, Math. Ann. 45, 278-294 (1894) and \textit{F. Schwarz} in: Symbolic and Algebraic Computation, Proc. ACM-SIGSAM 1989 Intl. Sympos., Portland 1989, 17-25 (1989)] for successively finding the \(L_i\) has serious gaps; the author fills in some of them.