The behaviour of meromorphic solutions to differential
equations has been the subject of much study. Research has
concentrated on the value distribution of meromorphic solutions
and their rates of growth. The purpose of the present paper is
to show that a thorough search will yield a list of all meromorphic
solutions to a multi-parameter ordinary differential equation introduced
by Hayman. This equation does not appear to be integrable
for generic choices of the parameters so we do not find all solutions
—only those that are meromorphic. This is achieved by combining
Wiman-Valiron theory and local series analysis. Hayman conjectured
that all entire solutions of this equation are of finite order.
All meromorphic solutions of this equation are shown to be either
polynomials or entire functions of order one.