Rational Heuristics for Rational Solutions of Riccati Equations

We describe some new algorithm and heuristics for computing the polynomial and rational solutions of bounded degree of a class of ordinary di erential equations, which includes generalized Riccati equations. As a consequence, our methods can be used for factoring linear ordinary di erential equations. Since they generate systems of algebraic equations in at most n unknowns, where n is the order of the di erential equation, they are particulary e ective for rst-order Riccati equations, which is con rmed by experimental timings for that case. Combined with the Ulmer-Weil algorithm and a modular heuristic for guessing the degree bounds, our method yield a rational heuristic alternative to Kovacic's algorithm for solving second order linear ordinary di erential equations.