Factorization method for some inhomogeneous Lienard equations


  • Octavio Cornejo Perez Facultad de Ingenieria, Universidad Autonoma de Queretaro, Centro Universitario Cerro de las Campanas, 76010 Santiago de Queretaro
  • Stefan C Mancas Department of Mathematics, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114-3900 http://orcid.org/0000-0003-1175-6869
  • Haret-Codratian Rosu Barbus Instituto Potosino de Investigación Científica y Tecnológica, A.C. http://orcid.org/0000-0001-5909-1945
  • Carlos Rico Olvera Universidad Autonoma de Queretaro, Centro Universitario Cerro de las Campanas, 76010 Santiago de Queretaro




Factorization, Inhomogeneous, Li\'enard equation, Abel equation, Riccati equation


The factorization of inhomogeneous Li\'enard equations is performed showing that through the factorization conditions involved in the method one can obtain forcing terms for which closed-form solutions exist. Because of the reduction of order feature of factorization, the solutions are simultaneously solutions of first-order differential equations with polynomial nonlinearities. Several illustrative examples of such solutions are presented, generically having rational parts and consequently singularities.


M.~Lakshmanan and S.~Rajasekar,

Nonlinear Dynamics: Integrability, Chaos, and Patterns, Springer, Heidelberg 2003.

T.~Harko and S.-D.~Liang,

Exact solutions of the Li'enard and generalized Li'enard type ordinary nonlinear differential equations obtained by

deforming the phase space coordinates of the linear harmonic oscillator,

J. Eng. Math. {bf 98} (2016) 93-111.

B.~van der Pol and J.~van der Mark, Frequency demultiplication, Nature {bf 120} (1927) 363–364.

H.C.~Rosu and O. Cornejo-P'erez,

Supersymmetric pairing of kinks for polynomial nonlinearities,

Phys. Rev. E {bf 71} (2005) 046607.

O.~Cornejo-P'erez and H.C.~Rosu,

Nonlinear second order ode's: Factorizations and particular solutions,

Prog. Theor. Phys. {bf 114} (2006) 533-538.

H.C. Rosu, O. Cornejo-P'erez, M. P'erez-Maldonado, J.A. Belinch'on,

Extension of a factorization method of nonlinear second order ODE's with variable coefficients,

Rev. Mex. F'{i}s. {bf 63} (2017) 218-222.

D.S.~Wang and H.~Li,

Single and multi-solitary wave solutions to a class of nonlinear evolution equations,

J. Math. Anal. Appl. {bf 343} (2008) 273-298.

V.K.~Chandrasekar, M.~Senthilvelan, and M.~Lakshmanan,

New aspects of integrability of force-free Duffing–van der Pol oscillator and related nonlinear systems,

J. Phys. A: Math. Gen. {bf 37} (2004) 4527-4534.


A closed form solution to a special normal form of Riccati equation,

Advances in Pure Mathematics {bf 1} (2011) 295


Asymptotic solutions of inhomogeneous differential equations having a turning point,

Stud. Appl. Math. (2020) 1–37. %(2020) arXiv:2005.08152.




How to Cite

O. Cornejo Perez, S. C. Mancas, H.-C. Rosu Barbus, and C. Rico Olvera, “Factorization method for some inhomogeneous Lienard equations”, Rev. Mex. Fís., vol. 67, no. 3 May-Jun, pp. 443–446, May 2021.



07 Gravitation, Mathematical Physics and Field Theory