Rational limit cycles on Bernoulli and Riccati equations

In this paper we deal with Bernoulli equations dy∕dx=A(x)yn+B(x)y, where A(x) and B(x) are real polynomials with A(x)⁄≡0 and n≥3. We prove that these Bernoulli equations can have at most 2 rational limit cycles if n is odd and at most one rational limit cycle if n is even. We also provide examples of Bernoulli equations with these numbers of rational limit cycles. Moreover we deal with the Riccati equations dy∕dx=A0(x)+A1(x)y+A2(x)y2, where A0(x),A1(x),A2(x) are real polynomials with A2(x)⁄≡0. We prove that these Riccati equations can have at most 2 rational limit cycles. We also provide examples of Riccati equations with these numbers of rational limit cycles.