2-descent for second order linear differential equations
Let L be a second order linear ordinary differential equation with coefficients in C ( x ). The goal in this paper is to reduce L to an equation that is easier to solve. The starting point is an irreducible L , of order two, and the goal is to decide if L is projectively equivalent to another equation L that is defined over a subfield C ( f ) of C ( x ). This paper treats the case of 2-descent, which means reduction to a subfield with index [ C ( x ): C ( f )]=2. Although the mathematics has already been treated in other papers, a complete implementation could not be given because it involved a step for which we do not have a complete implementation. The contribution of this paper is to give an approach that is fully implementable. Examples illustrate that this algorithm is very useful for finding closed form solutions (2-descent, if it exists, reduces the number of true singularities from n to at most n /2 + 2).
differential equations closed form solutions descent