Finite-order entire solutions of a class of algebraic differential equations (Q2212354)

The following statement is proved in the paper. Let \(F=C(z),R=F\{y\},P\in R,L\) be a linear differential polynomial over \(F\) and \(u(z)\) be an entire function of finite order. If \(L(u)=uP(u)\), then \(P(u)\in F\), that is, \(u\) is a Picard-Vessiot element over \(F.\) In particular, if \(y\) divides \(P(y)\), then \(u=q\exp(f)\), where \(q,f\in C[z]\).