The Reflection Function And Integrability Equivalence Of Several First-order Differential Equations

Studying the properties of solutions of the differential system x'=(t,x)is very important not only for the development of differential equation theory but also for practical application value of the law of motion of these objects in the objective world.Many research results have been given when the differential system is autonomous system.But for the research findings of non-autonomous differential system are relatively limited.As we all know,for periodic time-varying systems,we can study with well-known Poincare and Lyapunov transformations[1-4].However,it is very difficult for us to find these transformations.Mironenko professor[5]created reflection function theories in the 1980s.We can get the Poincare mapping of the periodic time-varying system x' = X(t,x)by using the reflective function,and in virtue of it we can study the qualitative behavior of its solutions.We call two differential system types with the same reflection function equivalence,and the periodic solution of the periodic system is the same.Therefore,in order to study the properties of a type of complex non-autonomous differential systems,we only need to study the behavior of solutions of a simple system or autonomous system which is equivalent to that system.Mironenko studied the equivalence of differential systems ???and ???.He has obtained(2)and(1)equivalence if and only if(2)can be expressed as ???.(3)where F(t,x)is the reflection function of(1).But it is very difficult to find the reflection function for the general differential system.How to determine that(1)and(2)are equivalent when the reflection function is unknown?So in[7],Mironenko gives that if ?(t,x)satisfies???then ???is equivalent to(1),where ?(t)is the odd quantity function of t.From this we can derive ???is also equivalent to(1),where ?i(t)is the odd quantity function,?i(t,x)is the solution of(4).It can be seen that it is very important to determine the equivalence of the two differential systems by finding the solution ?(t,x)of(4),i.e.reflective integral.Belskil gave the structural form of the reflection integral of Riccati equation ???,Abel equation ??? and generalpolynomial equation ???of these equations.Veresovich[19],Varenikova[25]studied the criterion of a planar polynomial differential system equivalent to its linear partIn this paper,I mainly study the reflection integral and inverse integral factor of some first order non-autonomous rational fractional differential equations.Through which the first order differential equations of equivalence with these equations are established,and the integrability and their qualitative behavior of these equations are studied by using inverse integral factors.Secondly,I study the equivalence of two non-autonomous linear equations and give some criteria for judgment.In the third chapter of this article,I study a rational fractional equation with sufficient conditions for various types of reflection integrals and establish a class of differential equations equivalent to(7).At the same time,I discuss the inverse integral factor of the differential system the first integral and the qualitative behavior of its solution.In addition,I study the sufficient conditions for the reflection integral of the quadratic rational fractional equation with the second rational fractional form,and establish the differential equation of equivalence with(9).Then I discuss the inverse integral factor of the differential system integrability problems and the qualitative behavior of its solution.In the fourth chapter,I study the equivalence of two non-autonomous linear differential systems x'=A(t)and x' =B(t)x.At the same time,we give the necessary conditions for the coefficient matrix A(t),B(t)satisfying them when they are equivalent and the several criteria for their equivalence.In particular,I also discuss the characteristic properties of ?(t),C when ??? are equivalent(where ?(t)is a pure function,C is a constant matrix).