Group analysis of the second order linear differential equation with variable delay

Abstract

We perform group analysis of the second order linear differential equation with the most general variable time delay by developing a Lie type invariance condition using Taylorís theorem for a function of more than one variable. This condition is then used to obtain the symmetry algebra and make a complete group classification of this delay differential equation. We deduce certain compatibility conditions for the infinitesimals, which lead to an extension of the symmetry algebra. Finally, we obtain a change of variables leading the differential equation with variable delay to be reduced to a differential equation with constant delay.