Integrals of \(\nu\)-graded systems of ordinary differential equations
type: Item
instance of: scholarly article
title: Integrals of \(\nu\)-graded systems of ordinary differential equations
zbMATH Open document ID: 0617.12013
DOI: 10.1016/0022-4049(87)90044-2
author: Lawrence Goldman
published in: Journal of Pure and Applied Algebra
publication date: 1987-01-01T00:00:00Z
review text: The problem of finding integrals of a system of n first-order differential equations is discussed: \[ Dy_ i=f_ i(y)\quad (i=1,...,n)\quad,\quad y=(y_ 1,...,y_ n) \] where the \(f_ i's\) are infinite series of rational functions in y and K, the field of coefficients, is a differential field. The author obtains an algorithm which yields all integrales of a system S which are in \(F_{\nu}\), if \(f_ i(y)\in F_{\nu}\) and \(\nu (f_ i(y))\geq \nu (y_ i)\) \((i=1,...,n)\), where \(\nu\) is some usual valuation of \(F=K(y)\) over K and \(F_{\nu}\) is the completion of F with respect to the given valuation. He shows how to use this algorithm to find rational or exponential integrals on the several examples. Also the author generalizes a result of Bruns for the N-body problem and of R. Liouville for the Euler equations.
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 37-XX
Mathematics Subject Classification ID: 12J10
zbMATH DE Number: 4000065
zbMATH Keywords: integrals of system of first-order differential equations
zbMATH Keywords: infinite series of rational functions
zbMATH Keywords: differential field
zbMATH Keywords: algorithm
zbMATH Keywords: valuation
zbMATH Keywords: N-body problem
zbMATH Keywords: Euler equations
MaRDI profile type: Publication
full work available at URL: 0022-4049(87)90044-2
OpenAlex ID: W1976336182
cites work: Q5557227
cites work: The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point
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description: scientific article; zbMATH DE number 4000065