Rational first integrals of the liénard equations: The solution to the poincaré problem for the liénard equations

Llibre, JaumePessoa, Claudio [UNESP]Ribeiro, Jarne D.2022-04-292022-04-292021-01-01Anais da Academia Brasileira de Ciencias, v. 93, n. 4, 2021.1678-26900001-3765http://hdl.handle.net/11449/229568Poincaré in 1891 asked about the necessary and sufficient conditions in order to characterize when a polynomial differential system in the plane has a rational first integral. Here we solve this question for the class of Liénard differential equations ẍ + f (x)ẋ + x = 0, being f (x) a polynomial of arbitrary degree. As far as we know it is the first time that all rational first integrals of a relevant class of polynomial differential equations of arbitrary degree has been classified.engLiénard equationPoincaré problemPolinomial differential equationRational first integralRational first integrals of the liénard equations: The solution to the poincaré problem for the liénard equationsArtigo10.1590/0001-37652021201911392-s2.0-85115401790