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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 ..."
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Cited by 32 (10 self)
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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.
Linear Differential Equations and Products of Linear Forms
 Journal of Pure and Applied Algebra
, 1997
"... We show that liouvillian solutions of an nth order linear differential equation L(y) = 0 are related to semiinvariant forms of the differential Galois group of L(y) = 0 which factor into linear forms. The logarithmic derivative of such a form F , evaluated in the solutions of L(y) = 0, is the f ..."
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Cited by 23 (5 self)
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We show that liouvillian solutions of an nth order linear differential equation L(y) = 0 are related to semiinvariant forms of the differential Galois group of L(y) = 0 which factor into linear forms. The logarithmic derivative of such a form F , evaluated in the solutions of L(y) = 0, is the first coefficient of a polynomial P (u) whose zeros are logarithmic derivatives of solutions of L(y) = 0. Together with the Brill equations, this characterisation allows one to efficiently test if a semiinvariant corresponds to such a coefficient and to compute the other coefficients of P (u) via a factorization of the form F . 1 Introduction In this paper k is a differential field whose field of constants C is algebraically closed of characteristic 0 (e.g. Q(x) with the usual derivation d=dx). For the derivation ffi of k and a 2 k we write ffi k (a) = a (k) and also a (1) = a 0 , a (2) = a 00 ; : : : . Let L(y) = a n d n y dx n + a n\Gamma1 d n\Gamma1 y dx n\Gamma1 + ...
On Second Order Homogeneous Linear Differential Equations with Liouvillian Solutions
 THEOR. COMP. SCIENCE
, 1997
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Computer Algebra Algorithms for Linear Ordinary Differential and Difference equations
 In Proceedings of the third European congress of mathematics
, 2001
"... Galois theory has now produced algorithms for solving linear ordinary differential and difference equations in closed form. In addition, recent algorithmic advances have made those algorithms effective and implementable in computer algebra systems. After introducing the relevant parts of the theory, ..."
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Cited by 7 (2 self)
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Galois theory has now produced algorithms for solving linear ordinary differential and difference equations in closed form. In addition, recent algorithmic advances have made those algorithms effective and implementable in computer algebra systems. After introducing the relevant parts of the theory, we describe the latest algorithms for solving such equations.
Liouvillian solutions of third order differential equations
 SUBMITTED TO JOURNAL OF SYMBOLIC COMPUTATION
"... The Kovacic algorithm and its improvements give explicit formulae for the Liouvillian solutions of second order linear differential equations. Algorithms for third order differential equations also exist, but the tools they use are more sophisticated and the computations more involved. In this paper ..."
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Cited by 7 (0 self)
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The Kovacic algorithm and its improvements give explicit formulae for the Liouvillian solutions of second order linear differential equations. Algorithms for third order differential equations also exist, but the tools they use are more sophisticated and the computations more involved. In this paper we refine parts of the algorithm to find Liouvillian solutions of third order equations. We show that, except for 4 finite groups and a reduction to the second order case, it is possible to give a formula in the imprimitive case. We also give necessary conditions and several simplifications for the computation of the minimal polynomial for the remaining finite set of finite groups (or any known finite group) by extracting ramification information from the character table. Several examples have been constructed, illustrating the possibilities and limitations.
A Kovacicstyle algorithm for liouvillian solutions of third order differential equations
"... The Kovacic algorithm [7] and its improvements (e.g. [18]) give explicit formulas for the Liouvillian solutions of second order linear differential equations. In this paper we extend this approach to third order equations. The result will either be a formula or a proof that the differential Galois g ..."
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Cited by 1 (0 self)
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The Kovacic algorithm [7] and its improvements (e.g. [18]) give explicit formulas for the Liouvillian solutions of second order linear differential equations. In this paper we extend this approach to third order equations. The result will either be a formula or a proof that the differential Galois group is one of 12 finite groups (it is well known how to solve a differential equation with known finite differential Galois group [3, 14, 17]). The approach uses only linear algebra, factorization of differential operators and computation of exponential solutions (all available in Maple), making the result easier to understand and to use for non specialists than [6]. The algorithm also decides if a third order differential equation which cannot be solved in terms of Liouvillian solutions can be solved using solutions of second order equations. AMS classification: 12H05, 68Q40 1