Bronstein, Manuel On solutions of linear ordinary differential equations in their coefficient field. (English) Zbl 0752.34009 J. Symb. Comput. 13, No. 4, 413-439 (1992). Summary: We describe a rational algorithm for finding the denominator of any solution of a linear ordinary differential equation in its coefficient field. As a consequence, there is now a rational algorithm for finding all such solutions when the coefficients can be built up from the rational functions by finitely many algebraic and primitive adjunctions. This also eliminates one of the computational bottlenecks in algorithms that either factor or search for Liouvillian solutions of such equations with Liouvillian coefficients. Cited in 1 ReviewCited in 13 Documents MSC: 65J99 Numerical analysis in abstract spaces 68W30 Symbolic computation and algebraic computation 12H20 Abstract differential equations 34A30 Linear ordinary differential equations and systems 65L99 Numerical methods for ordinary differential equations Keywords:rational algorithm; linear ordinary differential equation; coefficient field; Liouvillian solutions; Liouvillian coefficients PDFBibTeX XMLCite \textit{M. Bronstein}, J. Symb. Comput. 13, No. 4, 413--439 (1992; Zbl 0752.34009) Full Text: DOI References: [1] Abramov, S. A., Rational Solutions of Linear Differential and Difference Equations with Polynomial Coefficients (in russian), Journal of Computational Mathematics and Mathematical Physics, 29, No.11, 1611-1620 (1989) · Zbl 0695.65051 [2] Bronstein, M., Integration of Elementary Functions, Journal of Symbolic Computation, 9, No. 2, 117-173 (1990) · Zbl 0718.12006 [3] Bronstein, M., A Unification of Liouvillian Extensions, Applicable Algebra in Engineering, Communication and Computing, 1, No.1, 5-24 (1990) · Zbl 0757.12003 [4] Bronstein, M., The Risch Differential Equation on an Algebraic Curve, (Proceedings of ISSAC ’91 (1991), ACM Press: ACM Press New York), 241-246 · Zbl 1019.12500 [5] Davenport, J. H., Intégration Algorithmique des Fonctions Elémentairement Transcendantes sur une Courbe Algébrique, Annales de l’Institut Fourier, 34, 271-276 (1984), fasc.2 · Zbl 0506.34002 [6] Risch, R., On the Integration of Elementary Functions which are built up using Algebraic Operations, (Report SP-2801/002/00 (1968), System Development Corp.: System Development Corp. Santa Monica, CA) [7] Schwarz, F., A Factorization Algorithm for Linear Ordinary Differential Equations, (Proceedings of ISSAC ’89 (1989), ACM Press: ACM Press New York), 17-25 [8] Singer, M. F., Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients, Journal of Symbolic Computation, 11, 251-273 (1991) · Zbl 0776.12002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.