The Geometry of Fourth Order Differential Operator

In present paper, the equivalence problem for fourth order differential operators with one variable under general fiber-preserving transformation using the Cartan method of equivalence is applied. Two versions of equivalence problems are considered. First, the direct equivalence problem and second equivalence problem is to determine the sufficient and necessary conditions on two fourth order differential operators such that there exists a fiber-preserving transformation mapping one to the other according to gauge equivalence.