Computation of all rational solutions of first-order algebraic ODEs

In this paper, we discuss three different approaches to attack the problem of determining all rational solutions for a first-order algebraic ordinary differential equation (AODE). We first give a sufficient condition for first-order AODEs to have the property that poles of rational solutions can only occur at the zeros of the leading coefficient. A combinatorial approach is presented to determine all rational solutions, if there are any, of the family of first-order AODEs satisfying this condition. Algebraic considerations based on algebraic function theory yield another algorithm for quasi-linear first-order AODEs. And finally ideas from algebraic geometry combine these results to an algorithm for finding all rational solutions of parametrizable first-order AODEs.