Cartan equivalence problem for third order differential operators

This article is dedicated to solve the equivalence problem for two third order differential operators on the line under general fiber–preserving transformation using the Cartan method of equivalence. We will do three versions of the equivalence problems: first via the direct equivalence problem, second equivalence problem is to determine conditions on two differential operators such that there exists a fiber-preserving transformations mapping one to the other according to gauge equivalence.