Ordinary differential equations invariant under translation in the independent variable and rescaling: The Lagrangian formulation

The Lagrangian formulation of the class of general second-order ordinary differential equations invariant under translation in the independent variable and rescaling is presented. The differential equations arising from this analysis are analysed using the Painlevé test. The well-known differential equation, y″+yy′+ky3=0, is a unique member of this class when k=3 since it is linearisable by a point transformation. A wider subset is shown to be linearisable by means of a nonlocal transformation.