We study ordinary algebraic differential equations. We clarified permissible operations in our paper “On the irreducibility of the first differential equation of Painlevé” [Alg. Geo. and Comm. Alg. in Honor of M. Nagata, Kinokuniya, Tokyo, 771-789 (1987)]. Roughly speaking, the permissible operations consist of solution of linear equations and substitution of known functions in Abelian functions. In this paper we give a rigorous proof for the following results whose original form is found in Painlevé’s Stokholm Lectures (1895):
(1) An algebraic differential equation whose general solution depends rationally on the initial condition is solvable by permissible operations;
(2) An algebraic differential equation of the second order free from movable singular points is irreducible if and only if the general solution depends essentially transcendentally on the initial conditions.
As an application, we give the second proof of the irreducibility of the first differential equation \(y''=6y^ 2+x\) of Painlevé.