An algorithm for solving second order linear homogeneous differential equations
Journal of Symbolic Computation
Liouvillian and algebraic solutions of second and third order linear differential equations
Journal of Symbolic Computation
The MAGMA algebra system I: the user language
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference
Liouvillian solutions of linear differential equations of order three and higher
Journal of Symbolic Computation - Special issue on differential algebra and differential equations
Computing galois groups of linear differential equations of order four
Computing galois groups of linear differential equations of order four
Fourth order linear differential equations with imprimitive group
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Rational general solutions of algebraic ordinary differential equations
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Algebraic general solutions of algebraic ordinary differential equations
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Polynomial general solutions for first order autonomous ODEs
IWMM'04/GIAE'04 Proceedings of the 6th international conference on Computer Algebra and Geometric Algebra with Applications
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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.