# On the Integrability of the Abel and of the Extended Liénard Equations

On the Integrability of the Abel and of the Extended Liénard Equations We present some exact integrability cases of the extended Liénard 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 Liénard equation to an equivalent first kind first order Abel type equation given by $$\frac{d v}{d y}=f(y) v^{3-n}+k(y) v^{3-m}+g(y) v^{2}+h(y) v^{3}$$ d v d y = f ( y ) v 3 − n + k ( y ) v 3 − m + g ( y ) v 2 + h ( y ) v 3 , with υ = 1/y′. As a first step in our study we obtain three integrability cases of the extended quadratic-cubic Liénard 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 Liénard 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 Liénard 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 Liénard equation. The application of the generalized Chiellini condition to the case of the reduced Riccati equation is also considered. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# On the Integrability of the Abel and of the Extended Liénard Equations

, Volume 35 (4) – Dec 19, 2019
15 pages      /lp/springer-journals/on-the-integrability-of-the-abel-and-of-the-extended-li-nard-equations-mca4FKz7Zb
Publisher
Springer Journals
Copyright © 2019 by The Editorial Office of AMAS & Springer-Verlag GmbH Germany
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-019-0847-1
Publisher site
See Article on Publisher Site

### Abstract

We present some exact integrability cases of the extended Liénard 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 Liénard equation to an equivalent first kind first order Abel type equation given by $$\frac{d v}{d y}=f(y) v^{3-n}+k(y) v^{3-m}+g(y) v^{2}+h(y) v^{3}$$ d v d y = f ( y ) v 3 − n + k ( y ) v 3 − m + g ( y ) v 2 + h ( y ) v 3 , with υ = 1/y′. As a first step in our study we obtain three integrability cases of the extended quadratic-cubic Liénard 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 Liénard 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 Liénard 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 Liénard equation. The application of the generalized Chiellini condition to the case of the reduced Riccati equation is also considered.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Dec 19, 2019

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