An algorithm for solving second order linear homogeneous differential equations
Journal of Symbolic Computation
Galois groups of second and third order linear differential equations
Journal of Symbolic Computation
Liouvillian and algebraic solutions of second and third order linear differential equations
Journal of Symbolic Computation
Irreducible linear differential equations of prime order
Journal of Symbolic Computation
Galois action on solutions of a differential equation
Journal of Symbolic Computation
Journal of Symbolic Computation
On symmetric powers of differential operators
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Constructing a third-order linear differential equation
Theoretical Computer Science - Special volume on computer algebra
Liouvillian solutions of linear differential equations of order three and higher
Journal of Symbolic Computation - Special issue on differential algebra and differential equations
Linear differential operators for polynomial equations
Journal of Symbolic Computation
Solving third order linear differential equations in terms of second order equations
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Solving difference equations whose coefficients are not transcendental
Theoretical Computer Science
Hi-index | 0.00 |
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 four 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.