Solutions of first order differential equations which are solutions of linear differential equations of higher order

Abstract

Let 5f be an ordinary differential field (i.e., a field with a given derivation) of characteristic zero. An element z belonging to a differential field extension of 5f is said to be of order r over 5f if the lowest order irreducible differential equation with coefficients in 5f that z satisfies is of order r. It follows that z is of order r over 5f if and only if the degree of transcendency of i3(z) over 5f is r. In [4] Ritt proved that if P, QEE{y} and Q vanishes for every zero of P then the sum of the lowest (highest) degree terms of Q vanishes for every zero of the sum of the lowest (highest) degree terms of P. For our purpose we need the following slight generalization; it can be obtained either by essentially the same proof as used by Ritt, or as an almost immediate corollary of this theorem.