摘要
This paper considers algebraic ordinary differential equations(AODEs)and study their polynomial and rational solutions.The authors first prove a sufficient condition for the existence of a bound on the degree of the possible polynomial solutions to an AODE.An AODE satisfying this condition is called noncritical.Then the authors prove that some common classes of low-order AODEs are noncritical.For rational solutions,the authors determine a class of AODEs,which are called maximally comparable,such that the possible poles of any rational solutions are recognizable from their coefficients.This generalizes the well-known fact that any pole of rational solutions to a linear ODE is contained in the set of zeros of its leading coefficient.Finally,the authors develop an algorithm to compute all rational solutions of certain maximally comparable AODEs,which is applicable to 78.54%of the AODEs in Kamke's collection of standard differential equations.
基金
supported by Vietnam National Foundation for Science and Technology Development(NAFOSTED)under Grant No.101.04-2019.06
supported by the Austrian Science Fund(FWF)under Grant No.P29467-N32
the UTD startup Fund under Grant No.P-1-03246
the Natural Science Foundations of USA under Grant No.CF-1815108 and CCF-1708884。