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

The Cartan equivalence method is utilized to deduce an invariant characterization of the scalar third-order ordinary differential equation $u^{\mathrm{\prime }}\mathrm{"}=f(x,u,u^{\mathrm{\prime }},u\mathrm{"})$ 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 ${\bar{u}}^{\mathrm{\prime }}\mathrm{"}=0$. Moreover, examples are given to illustrate the method.