Über eine Klasse von Painlevéschen Differentialgleichungen. (On a class of differential equations of Painlevé type)
type: Item
instance of: scholarly article
title: Über eine Klasse von Painlevéschen Differentialgleichungen. (On a class of differential equations of Painlevé type)
zbMATH Open document ID: 0601.34005
author: Norbert Steinmetz
published in: Arkiv för Matematik
publication date: 1983-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 34M55
zbMATH DE Number: 3969079
zbMATH Keywords: second order differential equation
zbMATH Keywords: Riccati's equation
zbMATH Keywords: movable singularities
zbMATH Keywords: meromorphic-transcendental solutions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3969079