Über eine Klasse von Painlevéschen Differentialgleichungen. (On a class of differential equations of Painlevé type)
Among all differential equations of first order of the form \(w'=R(z,w)\) with rational R, Riccati's equation is exhibited from two aspects: its solutions do not have movable singularities, and it is the only type of equation having meromorphic-transcendental solutions (i.e. transcendental solutions with only poles as finite singularities). For second-order equations, to describe those having one of the above properties is much more difficult. The author gives a theorem which shows that among equations of the form \(w''=P(z,w)+Q(z,w)w'\) (P and Q polynomials in w with rational coefficients), those which have a meromorphic- transcendental solution are of a very special type. A second theorem is an improved version of the first one for a more restricted type of equation.
scientific article; zbMATH DE number 3969079