  In this paper, we carry out the equivalence problem for fifth-order
differential operators on the line under general fiber-preserving
transformation using the Cartan method of equivalence. Two versions of
equivalence problems have been solved. We consider the direct equivalence
problem and an equivalence problem to determine the sufficient and necessary
conditions on fifth-order differential operators such that there exists a
fiber--preserving transformation mapping one to the other according to gauge
equivalence.
