Abstract
We consider the general Liénard-type equation \( "u = \sum\nolimits_{k = 0}^n {{f_k}\mathop {{u^k}}\limits^. } \) for n ≥ 4. This equation naturally admits the Lie symmetry \({\partial \over {\partial t}}\). We completely characterize when this equation admits another Lie symmetry, and give an easily verifiable condition for this on the functions f0 ,..., fn. Moreover, we give an equivalent characterization of this condition. Similar results have already been obtained previously in the cases n = 1 or n = 2. That is, this paper handles all remaining cases except for n = 3.
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Figula, Á., Horváth, G., Milkovszki, T. et al. The Lie symmetry group of the general Liénard-type equation. J Nonlinear Math Phys 27, 185–198 (2020). https://doi.org/10.1080/14029251.2020.1700623
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DOI: https://doi.org/10.1080/14029251.2020.1700623
Key words
- second order ordinary differential equation
- Liénard-type equation
- Levinson-Smith-type equation
- Lie group
- symmetry analysis
- Lie algebra of infinitesimal symmetries