On the first integral and equivalence of nonlinear differential equations

In this paper, we study the equivalence of several different types of differential systems and provide a new method to find out the first integral of such differential systems. We draw some sufficient conditions for a general time-varying nonlinear differential system to be equivalent to a given differential system. We use the obtained results to discuss the qualitative behavior of the periodic solutions of such time-varying differential systems and get some interesting conclusions.