# Lie point symmetries classification of the mixed Liénard-type equation

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## Abstract

In this paper we develop a systematic and self-consistent procedure based on a set of compatibility conditions for identifying all maximal (eight parameter) and non-maximal (one and two parameter) symmetry groups associated with the mixed quadratic-linear Liénard-type equation, \(\ddot{x} + f(x){\dot{x}}^{2} + g(x)\dot{x}+h(x)= 0\), where \(f(x),\,g(x)\) and *h*(*x*) are arbitrary functions of *x*. With the help of this procedure we show that a symmetry function *b*(*t*) is zero for non-maximal cases, whereas it is not so for the maximal case. On the basis of this result the symmetry analysis gets divided into two cases, (i) the maximal symmetry group \((b\ne 0)\) and (ii) non-maximal symmetry groups \((b=0)\). We then identify the most general form of the mixed quadratic linear Liénard-type equation in each of these cases. In the case of eight-parameter symmetry group, the identified general equation becomes linearizable. In the case of non-maximal symmetry groups the identified equations are all integrable. The integrability of all the equations is proved either by providing the general solution or by constructing time-independent Hamiltonians.

## Keywords

Lie point symmetries Liénard-type equation Ordinary differential equations## Notes

### Acknowledgments

AKT and SNP are grateful to the Centre for Nonlinear Dynamics, Bharathidasan University, Tiruchirappalli, for warm hospitality. The work of SNP forms part of a Department of Science and Technology, Government of India, sponsored research project. The work of MS forms part of a research project sponsored by UGC. The work forms part of a Department of Science and Technology, Government of India IRHPA project and a DST Ramanna Fellowship project of ML. He also acknowledges the financial support provided through a DAE Raja Ramanna Fellowship.

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