On the integrability of the Abel and of the extended Liénard equations

Abstract: We present some exact integrability cases of the extended Li\'{e}nard equation $y^{\prime \prime }+f\left( y\right) \left(y^{\prime }\right)^{n}+k\left( y\right) \left(y^{\prime }\right)^{m}+g\left(y\right) y^{\prime }+h\left( y\right) =0$, 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\'{e}nard equation to an equivalent first kind first order Abel type equation given by $\frac{dv}{dy} =f\left( y\right) v^{3-n}+k\left( y\right) v^{3-m}+g\left( y\right) v^{2}+h\left( y\right) v^{3}$, with $v=1/y^{\prime }$. As a first step in our study we obtain three integrability cases of the extended quadratic-cubic Li\'{e}nard 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\'{e}nard 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\'{e}nard equation can be obtained by quadratures. The Chiellini integrability condition is extended to generalized Abel equations with $g(y)\equiv 0$ and $h(y)\equiv 0$, and arbitrary $n$ and $m$, thus allowing to obtain the general solution of the corresponding Li\'{e}nard equation. The application of the generalized Chiellini condition to the case of the reduced Riccati equation is also considered.