TY - JOUR

T1 - Rational Solutions of High-Order Algebraic Ordinary Differential Equations

AU - Vo, Thieu N.

AU - Zhang, Yi

N1 - Publisher Copyright:
© 2019, The Editorial Office of JSSC & Springer-Verlag GmbH Germany.

PY - 2020/6/1

Y1 - 2020/6/1

N2 - 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.

AB - 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.

KW - Algebraic ordinary differential equations

KW - algorithms

KW - polynomial solutions

KW - rational solutions

UR - http://www.scopus.com/inward/record.url?scp=85077594360&partnerID=8YFLogxK

U2 - 10.1007/s11424-019-8133-0

DO - 10.1007/s11424-019-8133-0

M3 - Article

AN - SCOPUS:85077594360

SN - 1009-6124

VL - 33

SP - 821

EP - 835

JO - Journal of Systems Science and Complexity

JF - Journal of Systems Science and Complexity

IS - 3

ER -