Parametric conditions and exact solution for the Duffing-Van der Pol class of equations
Keywords:parametric conditions, Prelle and Singer method, Lie symmetry method, exact solutions, Duffing-Van der Pol equation, Riccati equation, hypergeometric functions
This paper presents a methodology to find the exact solution and respective parametric conditions to the Duffing-Van der Pol class of equations. The supposed method in this paper is different from the Prelle and Singer method and the Lie symmetry method. The main idea of the supposed method is implemented in finding the first integrals of the original equation and leading this equation to a solved equation of lower order to which the exact solution can be obtained. As results the parametric conditions and the exact solutions in parametric forms are indicated. The algorithm for determining integral constants and the investigation of solution characteristics are considered.
M. J. Prelle and M. F. Singer. Elementary first integrals of differential equations. Transactions of the American Mathematical Society, 279, (1), (1983), pp. 215–229. https://doi.org/10.1090/s0002-9947-1983-0704611-x
L. G. S. Duarte, S. E. S. Duarte, L. A. C. P. da Mota, and J. E. F. Skea. Solving second-order ordinary differential equations by extending the Prelle-Singer method. Journal of Physics A: Mathematical and General, 34, (14), (2001), pp. 3015–3024. https://doi.org/10.1088/0305-4470/34/14/308
V. K. Chandrasekar, M. Senthilvelan, and M. Lakshmanan. On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. 461. The Royal Society, (2005), pp. 2451–2477. https://doi.org/10.1098/rspa.2005.1465
M. S. Velan and H. Lakshmanan. Lie symmetries and infinite-dimensional Lie algebras of certain nonlinear dissipative systems. Journal of Physics A: Mathematical and General, 28, (7), (1995), pp. 1929–1942. https://doi.org/10.1088/0305-4470/28/7/015
V. K. Chandrasekar, M. Senthilvelan, and M. Lakshmanan. New aspects of integrability of force-free Duffing–van der Pol oscillator and related nonlinear systems. Journal of Physics A: Mathematical and General, 37, (16), (2004), pp. 4527–4534. https://doi.org/10.1088/0305-4470/37/16/004
D.W. Jordan and P. Smith. Nonlinear ordinary differential equations: an introduction to dynamical systems Oxford University Press, New York, (2007).
M. Lakshmanan and S. Rajaseekar. Nonlinear dynamics: integrability, chaos and patterns Springer Science & Business Media, (2012).
G. Gao and Z. Feng. First integrals for the Duffing–van der Pol type oscillator. Electronic Journal of Differential Equations, 19, (2010), pp. 123–133.
D. H. Bich and N. D. Bich. A coupling successive approximation method for solving Duffing equation and its application. Vietnam Journal of Mechanics, 36, (2), (2014), pp. 77–93. https://doi.org/10.15625/0866-7136/36/2/4095
Wolfram Mathematica Tutorial Collection. Mathematics and Algorithms, Wolfram Research, Inc., United States of America, (2008).