On Liouvillian Solutions of Linear Differential Equations of Order 4 and 5 (2001)
| Venue: | Distributed Artificial Intelligence |
| Citations: | 7 - 0 self |
BibTeX
@INPROCEEDINGS{Cormier01onliouvillian,
author = {Olivier Cormier and F- Rennes Cedex},
title = {On Liouvillian Solutions of Linear Differential Equations of Order 4 and 5},
booktitle = {Distributed Artificial Intelligence},
year = {2001},
publisher = {Press}
}
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Abstract
In this paper we give the minimal possible degrees of an algebraic solution of the Riccati equation associated to an irreducible linear homogeneous differential equation of order 4 and 5, following the method used for the order 3 ([14]). We show that the algebraic degree of such a solution is bounded by 120 for the order 4 and by 55 for the order 5. With the important work done by Hessinger in [5] and the Tables computed in the sequel this leads to an algorithm to find Galois group and liouvillian solutions of equations of order 4. In the last section we construct an irreducible differential equation of degree 4 having SL(2; 7) as a Galois group and compute its liouvillian solutions. We also give the minimal polynomial over Q(x) of the algebraic solutions of the differential equation. This example illustrates a method which is easy to use in order to compute a liouvillian solution with the help of the Tables computed in this paper. 1.
Keyphrases
liouvillian solution linear differential equation galois group algebraic solution minimal polynomial last section differential equation minimal possible degree important work irreducible differential equation irreducible linear homogeneous differential equation riccati equation algebraic degree
