Invariant characterization of scalar third-order ODEs that admit the maximal point symmetry Lie algebra

Abstract

The Cartan equivalence method is utilized to deduce an invariant characterization of the scalar third-order ordinary differential equation $u^{\prime }"=f(x,u,u^{\prime },u")$ which admits the maximal seven-dimensional point symmetry Lie algebra. The method provides auxiliary functions which can be used to efficiently obtain the point transformation that does the reduction to the simplest linear equation ${\bar{u}}^{\prime }"=0$. Moreover, examples are given to illustrate the method.