Nonlinear ordinary differential equations resolvable with respect to an irregular singular point (Q1327090)

The main subject of this paper is a differential equation \(x^{- r+1}{d\over dx} y(x)= f(x,y)\) which has the irregular singular point \(x=\infty\) of the Poincaré rank \(r\geq 1\). The conditions on \(f(x,y)\) is so lengthy that they are omitted here. It is proved that a resolvable system has a unique formal series solution in some fractional powers of \(x^{-1}\). Moreover, any system has a formal series solution if and only if it can be reduced to a resolvable one. The second Painlevé equation is considered as an example.