Rational Solutions of High-Order Algebraic Ordinary Differential Equations

Thieu N. Vo, Yi Zhang*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)821-835
Number of pages15
JournalJournal of Systems Science and Complexity
Issue number3
Publication statusPublished - 1 Jun 2020
Externally publishedYes


  • Algebraic ordinary differential equations
  • algorithms
  • polynomial solutions
  • rational solutions

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