Summary: Algorithms for Special Integrals of Ordinary Differential
David W Albrechty, Elizabeth L Mansfieldz and Alice E Milnex
yDepartment of Computer Science, Monash University, Clayton 3168 Australia
zInstitute of Mathematics and Statistics, University of Kent, Canterbury CT2 7NF, UK
xDepartment of Mathematics, University of Exeter, Exeter EX4 4QE, UK
Short title: Algorithms for Special Integrals
Submitted to: J. Phys. A: Math. Gen.
Date: 8 July 1998
We give new, efficient and simple procedures for calculating special integrals of polynomial type
(also known as Darboux polynomials, algebraic invariant curves, or eigenpolynomials), for ordinary
differential equations. In principle, the method requires only that the given ordinary differential
equation be itself of polynomial type of degree one and any order. The method is algorithmic, is
suited to the use of computer algebra, and does not involve solving large nonlinear algebraic systems.
To illustrate the method, special integrals of the second, fourth and sixth Painlev'e equations, and
a third order ordinary differential equation of Painlev'etype are investigated. We prove that for
the second Painlev'e equation, the known special integrals are the only ones possible.