Laurent Series Solutions of Algebraic Ordinary Differential Equations

This paper concerns Laurent series solutions of algebraic ordinary differential equations (AODEs). We first present several approaches to compute formal power series solutions of a given AODE. Then we determine a bound for orders of its Laurent series solutions. Using the order bound, one can transform a given AODE into a new one whose Laurent series solutions are only formal power series. The idea is basically inherited from the Frobenious method for linear ordinary differential equations. As applications, new algorithms are presented for determining all particular polynomial and rational solutions of certain classes of AODEs.