Abstract In this paper we deal with Bernoulli equations d y ∕ d x = A ( x ) y n + 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 d y ∕ d x = A 0 ( x ) + A 1 ( x ) y + A 2 ( x ) y 2 , where A 0 ( x ) , A 1 ( x ) , A 2 ( x ) are real polynomials with A 2 ( 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.

中文翻译：

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Bernouilli 和 Riccati 方程的有理极限环

摘要 本文讨论伯努利方程dy ∕ dx = A ( x ) yn + B ( x ) y ，其中A ( x ) 和B ( x ) 是实数多项式，其中A ( x ) ⁄ ≡ 0 且n ≥ 3 . 我们证明了这些伯努利方程如果 n 是奇数，最多可以有 2 个有理极限环，如果 n 是偶数，则最多可以有一个有理极限环。我们还提供了具有这些有理极限环数的伯努利方程的示例。此外，我们处理 Riccati 方程 dy ∕ dx = A 0 ( x ) + A 1 ( x ) y + A 2 ( x ) y 2 ，其中 A 0 ( x ) , A 1 ( x ) , A 2 ( x )是具有 A 2 ( x ) ⁄ ≡ 0 的实数多项式。我们证明这些 Riccati 方程最多可以有 2 个有理极限环。我们还提供了具有这些有理极限环数的 Riccati 方程的示例。