Rational Periodic Solutions on Some Generalized Abel Equations

Abstract

In this paper we deal with the equations a(x)dy/dx = A(x)y 2 + B(x)y 3, where a(x), A(x) and B(x) are complex polynomials with a(x)B(x) ≢ 0 and a(x) non-constant. First we show that the unique rational limit cycles that these equations can have are of the form y = 1/p(x) being p(x) some polynomial. Second we provide an upper bound on the number of these rational limit cycles. Moreover, we prove that if deg(B(x)) − deg(a(x)) + 1 is odd, or deg(A) > (deg(B(x)) + deg(a(x)) − 1)/2, then these Abel equations have at most two rational limit cycles and we provide examples of these Abel equations with three nontrivial rational periodic solutions.