New Symmetries for Ordinary Differential Equations

In this paper we present a theory for calculating new symmetries for ordinary differential equations. These new symmetries lead to a systematic reduction of the order of a differential equation. Our approach depends on computing the Lie symmetries of a differential equation related to a given differential equation by a Bäcklund transformation. Consequently we induce new symmetries of the given differential equation which are not in general of Lie, contact, or Lie-Bäcklund type. We obtain new symmetries and corresponding new analytic results for a class of ordinary differential equations arising from nonlinear diffusion.