Lie algebra classification, conservation laws and invariant solutions for the a particular case of the generalized Levinson-Smith equation

Abstract

In this study, we examine a specific instance of the generalized Levinson-Smith equation, which is linked to the Liènard equation and holds significant importance from the perspectives of physics, mathematics, and engineering. This underlying equation has practical applications in mechanics and nonlinear dynamics and has been extensively explored in the qualitative scheme. Our approach involves applying the Lie group method to this equation. By doing so, we derive the optimal generating operators for the system that pertain to the specific instance of the generalized Levinson-Smith equation. These operators are then used to define all invariant solutions associated with the equation. In addition, we demonstrate the variational symmetries and corresponding conservation laws using Noether's theorem. Finally, we categorize the Lie algebra related to the given equation.