A polynomial time algorithm for finding rational general solutions of first order autonomous ODEs

Abstract

We give a necessary and sufficient condition for an algebraic ODE to have a rational type general solution. For a first order autonomous ODE F = 0 , we give an exact degree bound for its rational solutions, based on the connection between rational solutions of F = 0 and rational parametrizations of the plane algebraic curve defined by F = 0 . For a first order autonomous ODE, we further give a polynomial time algorithm for computing a rational general solution if it exists based on the computation of Laurent series solutions and Padé approximants. Experimental results show that the algorithm is quite efficient.