The transcendental Risch differential equation
type: Item
instance of: scholarly article
title: The transcendental Risch differential equation
zbMATH Open document ID: 0707.12003
DOI: 10.1016/S0747-7171(08)80006-5
author: Manuel Bronstein
published in: Journal of Symbolic Computation
publication date: 1990-01-01T00:00:00Z
review text: Let \(k\) be a differential field of characteristic \(0\) and \(K\) be a differential field extension of \(k\) (with derivation \(x\mapsto x': K\to K\)); let \(\theta\) be an element of \(K\) satisfying the following conditions: \(\theta\) is transcendental over \(k\), \(k\) and \(k(\theta)\) have the same field of constants, and either \(\theta\) is primitive over \(k\) (i.e. \(\theta '\in k)\) or \(\theta\) is exponential over \(k\) (i.e. \(\theta '/\theta =\eta '\) for some \(\eta\in k\)). A differential equation \[ y'+fy=g\text{ with } f,g\in k(\theta) \tag{*} \] is called a Risch differential equation. There is a considerable interest in solving such a differential equation, because algorithms for the integration of elementary functions in finite terms rely on the resolution of Risch differential equations. In 1969 \textit{R. H. Risch} published an algorithm for solving (*) [cf. Trans. Am. Math. Soc. 139, 167--189 (1969; Zbl 0184.06702)] which required factoring the denominators of \(f\) and \(g\). Later, \textit{M. Rothstein} [Ph. D. Thesis, Univ. of Wisconsin, Madison, 1976] and \textit{J. H. Davenport} [SIAM J. Comput. 15, 903--918 (1986; Zbl 0632.65091)] published algorithms for solving (*) which use square-free decompositions of the denominators of \(f\) and \(g\). Davenport's algorithm has been implemented in the computer algebra systems Scratchpad II and Maple; however it requires \(f\) to be in a certain form (called weakly normalized), but no complete algorithm that makes \(f\) weakly normalized has been published. The author describes a new algorithm for solving equation (*) or for showing that it doesn't have a solution. This algorithm requires \(f\) to have an integral which is elementary over \(k(\theta)\) and reduces the problem to the same problem for an equation \(AQ'+BQ=C\) with elements \(A,B,C\) of \(k[\theta]\) if \(\theta\) is primitive over \(k\), resp. with elements \(A\in k[\theta]\) and \(B,C\in k[\theta,\theta^{-1}]\) if \(\theta\) is …
Mathematics Subject Classification ID: 12H20
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 12Y05
zbMATH DE Number: 4160893
zbMATH Keywords: integration in finite terms
zbMATH Keywords: differential field
zbMATH Keywords: Risch differential equation
zbMATH Keywords: algorithm
zbMATH Keywords: Rothstein's algorithm
zbMATH Keywords: Scratchpad II
Wikidata QID: Q115338685
reviewed by: Friedrich Schwarz
describes a project that uses: Maple
describes a project that uses: IBM Scratchpad
MaRDI profile type: Publication
cites work: Integration of elementary functions
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cites work: The Risch Differential Equation Problem
cites work: Q3790647
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cites work: The Problem of Integration in Finite Terms
cites work: Integration in Finite Terms
description: scientific article; zbMATH DE number 4160893