Lie symmetries of first order neutral differential equations

Abstract

In this paper we extend the method of obtaining symmetries of ordinary differ ential equations to first order non-homogeneous neutral differential equations with variable coefficients. The existing method for delay differential equations uses a Lie-Backlund ¨operator and an Invariant Manifold Theorem to define the operators which are used to obtainthe infinitesimal generators of the Lie group. In this paper, we adopt a different approach anduse Taylor’s theorem to obtain a Lie type invariance condition and the determining equationsfor a neutral differential equation. We then split this equation in a manner similar to thatof ordinary differential equations to obtain an over-determined system of partial differentialequations. These equations are then solved to obtain corresponding infinitesimals, and hence desired equivalent symmetries. We then obtain the symmetry algebra admitted by this neutraldifferential equation.