Results 1 
7 of
7
Liouvillian solutions of linear differential equations of order three and higher
 JOURNAL OF SYMBOLIC COMPUTATION
, 1999
"... Singer and Ulmer (1997) gave an algorithm to compute Liouvillian (“closedform”) solutions of homogeneous linear differential equations. However, there were several efficiency problems that made computations often not practical. In this paper we address these problems. We extend the algorithm in van ..."
Abstract

Cited by 32 (10 self)
 Add to MetaCart
Singer and Ulmer (1997) gave an algorithm to compute Liouvillian (“closedform”) solutions of homogeneous linear differential equations. However, there were several efficiency problems that made computations often not practical. In this paper we address these problems. We extend the algorithm in van Hoeij and Weil (1997) to compute semiinvariants and a theorem in Singer and Ulmer (1997) in such a way that, by computing one semiinvariant that factors into linear forms, one gets all coefficients of the minimal polynomial of an algebraic solution of the Riccati equation, instead of only one coefficient. These coefficients come “for free ” as a byproduct of our algorithm for computing semiinvariants. We specifically detail the algorithm in the cases of equations of order three (order two equations are handled by the algorithm of Kovacic (1986), see also Ulmer and Weil (1996) or Fakler (1997)). In the appendix, we present several methods to decide when a multivariate polynomial depending on parameters can admit linear factors, which is a necessary ingredient in the algorithm.
Solving second order linear differential equations with Klein’s theorem
 In ISSAC’05
, 2005
"... Given a second order linear differential equations with coefficients in a field k = C(x), the Kovacic algorithm finds all Liouvillian solutions, that is, solutions that one can write in terms of exponentials, logarithms, integration symbols, algebraic extensions, and combinations thereof. A theorem ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
(Show Context)
Given a second order linear differential equations with coefficients in a field k = C(x), the Kovacic algorithm finds all Liouvillian solutions, that is, solutions that one can write in terms of exponentials, logarithms, integration symbols, algebraic extensions, and combinations thereof. A theorem of Klein states that, in the most interesting cases of the Kovacic algorithm (i.e when the projective differential Galois group is finite), the differential equation must be a pullback (a change of variable) of a standard hypergeometric equation. This provides a way to represent solutions of the differential equation in a more compact way than the format provided by the Kovacic algorithm. Formulas to make Klein’s theorem effective were given in [4, 2, 3]. In this paper we will give a simple algorithm based on such formulas. To make the algorithm more easy to implement for various differential fields k, we will give a variation on the earlier formulas, namely we will base the formulas on invariants of the differential Galois group instead of semiinvariants. 1.
Some methods to solve linear differential equations in closed form
"... In this article, we review various methods to find closed form solutions of linear differential equations that can for example be found in the symbolic computation package maple. We focus on the presentation of the methods and not on the underlying mathematical theory which is differential Galois t ..."
Abstract
 Add to MetaCart
In this article, we review various methods to find closed form solutions of linear differential equations that can for example be found in the symbolic computation package maple. We focus on the presentation of the methods and not on the underlying mathematical theory which is differential Galois theory (see [Sin07, vdPS03]). The text contains many examples which have been computed using maple†. We motivate the search of solution by an integration problem and all the algorithms are motivated via this integration problem. 0.1.1 Integrating via linear differential equations In 18323, Joseph Liouville publishes two articles on the determination of integrals which are algebraic functions [Lio33]. His goal is to to design an algorithm which decides when the integral of an algebraic function is algebraic (and compute it when it is) or prove that the integral is not algebraic. In this
Solving Second Order Linear Differential Equations with Klein’s Theorem
"... Given a second order linear differential equations with coefficients in a field k = C(x), the Kovacic algorithm finds all Liouvillian solutions, that is, solutions that one can write in terms of exponentials, logarithms, integration symbols, algebraic extensions, and combinations thereof. A theorem ..."
Abstract
 Add to MetaCart
(Show Context)
Given a second order linear differential equations with coefficients in a field k = C(x), the Kovacic algorithm finds all Liouvillian solutions, that is, solutions that one can write in terms of exponentials, logarithms, integration symbols, algebraic extensions, and combinations thereof. A theorem of Klein states that, in the most interesting cases of the Kovacic algorithm (i.e when the projective differential Galois group is finite), the differential equation must be a pullback (a change of variable) of a standard hypergeometric equation. This provides a way to represent solutions of the differential equation in a more compact way than the format provided by the Kovacic algorithm. Formulas to make Klein’s theorem effective were given in [4, 2, 3]. In this paper we will give a simple algorithm based on such formulas. To make the algorithm more easy to implement for various differential fields k, we will give a variation on the earlier formulas, namely we will base the formulas on invariants of the differential Galois group instead of semiinvariants.
Algorithms for Solving Linear Ordinary Differential Equations
, 1997
"... This article presents the new domain of linear ordinary differential operators and shows how it works in a few examples. Furthermore, in a very informal way the algebraic point of view dealing with ordinary differential equations will be introduced. Using such tools allows to develop general algorit ..."
Abstract
 Add to MetaCart
(Show Context)
This article presents the new domain of linear ordinary differential operators and shows how it works in a few examples. Furthermore, in a very informal way the algebraic point of view dealing with ordinary differential equations will be introduced. Using such tools allows to develop general algorithms for solving linear equations. Introduction