On solutions of polynomial growth of ordinary differential equations
type: Item
instance of: scholarly article
title: On solutions of polynomial growth of ordinary differential equations
zbMATH Open document ID: 0699.34059
DOI: 10.1016/0022-0396(89)90129-0
author: I. P. Van den Berg
published in: Journal of Differential Equations
publication date: 1989-01-01T00:00:00Z
review text: In this extensive paper the author presents a theorem (main theorem) on the existence of solutions of polynomial growth of ordinary differential equations of type \(E:dY/dX=F(X,Y)\), where F is of class \(C^ 1\). The asymptotic behaviour of these solutions and the variation of neighbouring solutions are obtained by solving an asymptotic functional equation related to E. The proof of the main theorem uses methods of nonstandard analysis. By a linear, nonstandard change of variables, called a macroscope, the original equation is transformed into a slow-fast differential equation, or singular perturbation. The existence is then proved using nonstandard perturbation theory. The result is an extension of results by F. and M. Diener and G. Reeb on solutions of polynomial growth of rational differential equations.
Mathematics Subject Classification ID: 34E05
zbMATH DE Number: 4146471
zbMATH Keywords: solutions of polynomial growth
zbMATH Keywords: rational differential equations
reviewed by: Jan Ohriska
MaRDI profile type: Publication
full work available at URL: 0022-0396(89)90129-0
OpenAlex ID: W1589876526
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description: scientific article; zbMATH DE number 4146471