Algorithm for finding reduction transformation that transforms a rational ODE to solvable structure

The integrability problem of rational first order ODE y'=M(x,y)/N(x,y), M,N in R[x,y], which can be viewed as a planar vector field, has been a long term research interest in area of dynamical system, physics etc. Since the invention of the computer algebra system, many algorithms have been developed to find the explicit first integral of such ODE and Maple has the most advanced function, called dsolve to handle this problem. In this paper, we would like to introduce an efficient algorithm which can be viewed as a supplement of the existing dsolve command, to judge whether a given rational ODE is of "reducible" structure, that is, via a rational, non-linear algebraic transformation in terms of the variable y, the ODE can be reduced to a more simple structure y'=Σni=0 fi(x)yi. Many of the known solvable ODE, including Bernoulli, Riccati, Abel, Chini are of this structure, and the integrable type of such ODE can be classified using invariants under linear transformation. We have implemented this algorithm based on Maple. And the overall performance and efficiency of this algorithm is considerable.