Following the work of Fuchs in classifying first-order ordinary differential equations, Painlevé studied second-order ordinary differential equation of the form
 |
where 
is analytic in 
and rational in 
and 
. Painlevé found 50 types whose only movable singularities
are ordinary poles. This characteristic is known as the
Painlevé property. Six of the transcendents define new transcendents known
as Painlevé transcendents, and the
remaining 44 can be integrated in terms of classical transcendents, quadratures,
or the Painlevé transcendents.
