gauge equivalence problem on differential operators under fiber-preserving transformation

This paper focuses on investigating the equivalence problem for fifth-order differential operators (fodos) on the line under general fiber-preserving transformations. utilizing the cartan method of equivalence, the study specifically addresses the gauge equivalence problem, seeking to establish the conditions for two fodos to be related by a fiber-preserving transformation. by analyzing the properties of these operators, the research aims to identify conditions for their transformation while maintaining the fiber structure. the systematic approach of the cartan method is employed to derive the necessary conditions for gauge equivalence between these fodos. the study aims to enhance understanding of the equivalence problem for fodos and shed light on fiber-preserving transformations that uphold gauge equivalence.