Rational Solutions of High-Order Algebraic Ordinary Differential Equations

Abstract: We consider algebraic ordinary differential equations (AODEs) and study their polynomial and rational solutions. A sufficient condition for an AODE to have a degree bound for its polynomial solutions is presented. An AODE satisfying this condition is called \emph{noncritical}. We prove that usual low order classes of AODEs are noncritical. For rational solutions, we determine a class of AODEs, which are called \emph{maximally comparable}, such that the poles of their rational solutions are recognizable from their coefficients. This generalizes a fact from linear AODEs, that the poles of their rational solutions are the zeros of the corresponding highest coefficient. An algorithm for determining all rational solutions, if there is any, of certain maximally comparable AODEs, which covers $78.54\%$ AODEs from a standard differential equations collection by Kamke, is presented.