On first integrals, conservation laws and reduction of classes of Emden and Liénard equations

Abstract

We present a general method to construct first integrals for some classes of the well known second-order ordinary differential equations, viz., the Emden and Liénard classes of equations. The approach does not require a knowledge of a Lagrangian but, rather, uses the ‘multiplier approach’ (Anco and Bluman in Eur J Appl Math 13:545–566, 2002; Eur J Appl Math 13:567–585, 2002). It is then shown how a study of the invariance properties and conservation laws are used to ‘twice’ reduce the equations to solutions. The equations admit five first integrals of which two are independent but the significance of the five are that they correspond to a five-dimensional algebra of Noether symmetries obtained without the need to construct a Lagrangian.