Documents related to the paper
« solving second order differential equations with Klein’s theorem »
by Mark van Hoeij and Jacques-Arthur Weil


Draft implementation of the pullback algorithm (in maple 9)

Maple commands for checking the results in the paper

A maple worksheet that shows how to prove and use the results

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 (obtained
by 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 semi-invariants.

This work takes its roots in previous work by Maint Berkenbosch and the authors.

Interested readers can read much more about pullbacks (rationality, more theory, order 3 equations, etc) and other topics (moduli spaces for differential equations) in Maint Berkenbosch's thesis (chapter 1 concerns pullbacks)

The results of our paper are (currently being) implemented also in:

Solve your differential equation on the web (Manuel Bronstein, projet CAFE, INRIA, Sophia-Antipolis)

Our Home Pages:
Mark van Hoeij
Jacques-Arthur Weil