Corpus ID: 237532415

# Rational Solutions of Abel Differential Equations

@inproceedings{Bravo2021RationalSO,
title={Rational Solutions of Abel Differential Equations},
author={Jennifer Bravo and Lucas A. Calder{\'o}n and M. Fernandez and Ignacio Ojeda},
year={2021}
}
We study the rational solutions of the Abel equation x′ = A(t)x + B(t)x where A,B ∈ C[t]. We prove that if deg(A) is even or deg(B) > (deg(A) − 1)/2 then the equation has at most two rational solutions. For any other case, an upper bound on the number of rational solutions is obtained. Moreover, we prove that if there are more than (deg(A) + 1)/2 rational solutions then the equation admits a Darboux first integral.

#### References

SHOWING 1-10 OF 25 REFERENCES
On the rational limit cycles of Abel equations
• Mathematics
• 2018
Abstract In this paper, we deal with Abel equations: d x d y = A ( x ) y 2 + B ( x ) y 3 , where A(x) and B(x) are real polynomials. If a solution y = φ ( x ) of the above equations satisfies that φExpand
On the Polynomial Solutions and Limit Cycles of Some Generalized Polynomial Ordinary Differential Equations
We study equations of the form y d y / d x = P ( x , y ) where P ( x , y ) ∈ R [ x , y ] with degree n in the y-variable. We prove that this ordinary differential equation has at most n polynomialExpand
On the number of polynomial solutions of Bernoulli and Abel polynomial differential equations
• Mathematics
• 2017
Abstract In this paper we determine the maximum number of polynomial solutions of Bernoulli differential equations and of some integrable polynomial Abel differential equations. As far as we know,Expand
The number of polynomial solutions of polynomial Riccati equations
• Mathematics
• 2016
Consider real or complex polynomial Riccati differential equations a(x) y=b_0(x) b_1(x)y b_2(x)y^2 with all the involved functions being polynomials of degree at most . We prove that the maximumExpand
On the Abel differential equations of third kind
• Mathematics
• 2020
Abel equations of the first and second kind have been widely studied, but one question that never has been addressed for the Abel polynomial differential systems is to understand the behavior of itsExpand
On the polynomial limit cycles of polynomial differential equations
• Mathematics
• 2011
In this paper we deal with ordinary differential equations of the form dy/dx = P(x, y) where P(x, y) is a real polynomial in the variables x and y, of degree n in the variable y. If y = φ(x) is aExpand
Center conditions at infinity for Abel differential equations
• Mathematics
• 2010
An Abel differential equation y' = p(x)y 2 + q(x)y 3 is said to have a center at a set A = {a 1 ,..., a r } of complex numbers if y(a 1 ) = y(a 2 ) =··· = y(a r ) for any solution y(x) (with theExpand
Rational limit cycles of Abel equations
• Mathematics
• 2021
We deal with Abel equations \begin{document}$dy/dx = A(x) y^2 + B(x) y^3$\end{document} , where \begin{document}$A(x)$\end{document} and \begin{document}$B(x)$\end{document} are realExpand
Polynomial Solutions of Equivariant Polynomial Abel Differential Equations
• Mathematics
• 2017
Abstract Let a ⁢ ( x ) {a(x)} be non-constant and let b j ⁢ ( x ) {b_{j}(x)} , for j = 0 , 1 , 2 , 3 {j=0,1,2,3} , be real or complex polynomials in the variable x. Then the real or complexExpand
Abel differential equations admitting a certain first integral
• Mathematics
• 2010
In this work algebro-geometric conditions to have a certain first integral for an Abel differential equation are given. These conditions establish a bridge with classical Galois theory because weExpand