Lie symmetries of first order neutral differential equations

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