An algorithm for the decomposition of differential polynomials in the general case

Abstract

The theory of decomposition of differential polynomials (DPs) is considered. For a given differential algebraic equation $P(x,y(x), y'(x), \ldots ,y^{(n)} (x)) = 0,$ the possibility to represent the differential polynomial P as the composition P = Q(x, R, R?, ..., R (q)), R = R(x, y(x), ..., y (r)(x)) of DPs Q and R is studied. It is shown that the decomposition problem is reduced to the factorization of linear ordinary differential operators (LODOs). A generic decomposition algorithm for DPs is described based on the Vandermonde differential theorem.