**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

**ABSTRACT**

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