On discrete groups and solvable equations of nonlinear oscillations

Abstract

The autonomous ordinary differential equation of Lienard is considered. This equation and its generalizations arise in the theory of nonlinear oscillations, dynamical systems, etc. Currently for equations of the Lienard type, the problem of finding exact solutions is relevant. In view of this, the problem is to find connections in Lienard type equations with investigated nonlinear equations. These problems are solved in this article for the Lienard equation with three-parametrical coefficients of polynomial type. We found a transformation that reduces the equation under consideration of the well-studied generalized Emden-Fowler equation. We indicated the method of finding exact solutions of Lienard equations with various parameters. The transformations discrete group of dihedron for the class of generalized Emden-Fauler equations induces a discrete group of transformations for the corresponding class of Lienard equations. Likewise, known solutions of some particular generalized Emden-Fauler equations induce solutions of the Lienard equations. As examples, we found exact solutions in elementary (including polynomials) and special functions of several Lienard equations with certain parameters.