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.
