Painlevé Property

Following the work of Fuchs in classifying first-order Ordinary Differential Equations, Painlevé studied second-order ODEs of the form

$${d^2y\over dx^2}=F(y',y,x),$$

where $F$ is Analytic in $x$ and rational in $y$ and $y'$. 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.

See also Painlevé Transcendents