Weiran L\"{u}, Qiuying Li, and Chungchun Yang China University of Petroleum, China University of Petroleum, China University of Petroleum

Abstract : In this paper, we consider the differential equation $$F'-Q_1=R{\rm e}^{\alpha}(F-Q_2),$$ where $Q_1$ and $Q_2$ are polynomials with $Q_1Q_2\not\equiv 0, R$ is a rational function and $\alpha$ is an entire function. We consider solutions of the form $F = f^n,$ where $f$ is an entire function and $n\geq 2$ is an integer, and we prove that if $f$ is a transcendental entire function, then $\frac{Q_1}{Q_2}$ is a polynomial and $f'=\frac{Q_1}{nQ_2}f.$ This theorem improves some known results and answers an open question raised in \cite{16}.

Keywords : transcendental entire solutions, differential equation, Nevanlinna theory