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The Cartan equivalence method is applied to provide an invariantcharacterization of the third-order ordinary differential equation$u'''=f(x,u,u',u'')$ which admits a five-dimensional point symmetry Liealgebra. The invariant characterization is given in terms of the function $f$in a compact form. A simple procedure to construct the equivalent canonicalform by use of an obtained constant invariant is also presented. We also showhow one obtains the point transformation that does the reduction to linearform. Moreover, some applications are provided.

Invariant characterization of third-order ODEs $u'''=f(x,u,u',u'')$ that admit a five-dimensional point symmetry Lie algebra