Mak, Man KwongMan KwongMakHarko, TiberiuTiberiuHarko2025-12-302025-12-3020191618-3932https://hdl.handle.net/20.500.12740/23751We present some exact integrability cases of the extended Li ' enard equation y + f(y)(y)n + k(y)(y)m + g(y)y + h(y) = 0, with n > 0 and m > 0 arbitrary constants, while f(y), k(y), g(y), and h(y) are arbitrary functions. The solutions are obtained by transforming the equation Li ' enard equation to an equivalent first kind first order Abel type equation given by dv dy = f(y)v3- n + k(y)v3- m + g(y)v2 + h(y)v3, with v = 1/y. As a first step in our study we obtain three integrability cases of the extended quadratic-cubic Li ' enard equation, corresponding to n = 2 and m = 3, by assuming that particular solutions of the associated Abel equation are known. Under this assumption the general solutions of the Abel and Li ' enard equations with coefficients satisfying some differential conditions can be obtained in an exact closed form. With the use of the Chiellini integrability condition, we show that if a particular solution of the Abel equation is known, the general solution of the extended quadratic cubic Li ' enard equation can be obtained by quadratures. The Chiellini integrability condition is extended to generalized Abel equations with g(y) = 0 and h(y) = 0, and arbitrary n and m, thus allowing to obtain the general solution of the corresponding Li ' enard equation. The application of the generalized Chiellini condition to the case of the reduced Riccati equation is also considered.Acceso AbiertoAbel equationLienard equationgeneral solutionsintegrability conditionshttp://dx.doi.org/10.1007/s10255-019-0847-1