Rational solutions of Abel trigonometric polynomial differential equations

Abstract

In this paper we deal with the trigonometric polynomial differential equations of the form Y′=A(θ)Y2+B(θ)Y3, where A and B are real trigonometric polynomials with B(θ)≢0. We first prove that these equations has at most two trigonometric polynomial solutions (which will always be constant solutions) and show that this upper bound is reached. Second we provide an upper bound on the number of rational trigonometric solutions that these equations can have and we show that under some conditions related with the degree of the trigonometric polynomials A(θ) and B(θ), the number of rational solutions is two.