Linearization of third-order ordinary differential equations u‴=f(x,u,u′,u″) via point transformations

The linearization problem for scalar third-order ordinary differential equations via point transformations was solved partially in the works of Al-Dweik et al by the use of the Cartan equivalence method. In order to solve this problem completely, the Cartan equivalence method is applied to provide an invariant characterization of the linearizable third-order ordinary differential equation (Formula presented.), which admits a four-dimensional point symmetry Lie algebra. The invariant characterization is given in terms of function f in a compact form. A simple procedure to construct the equivalent canonical form by use of an obtained invariant is also presented. The approach provides auxiliary functions, which can be effectively utilized to determine the point transformation that does the reduction to the equivalent canonical form. Furthermore, illustrations to the main theorem and applications are given.