Computing the Galois Group of Some Parameterized Linear Differential Equation of Order Two

We extend Kovacic’s algorithm to compute the differential Galois group of some second order parameterized linear differential equation. In the case where no Liouvillian solutions could be found, we give a necessary and sufficient condition for the integrability of the system. We give various examples of computation. Introduction Let us consider the linear differential equation ( ∂XY (X) ∂2 XY (X) ) = ( 0 1 r(X) 0 )( Y (X) ∂XY (X) ) , where r(X) is a rational function with coefficients in C. We have a Galois theory for this type of equation; see [VdPS]. In particular, we can associate to this equation a group H, which we call the differential Galois group, that measures the algebraic relations of the solutions. In this case, this group can be viewed as a linear algebraic subgroup of SL2(C). Kovacic in [Kov] (see also [VdP]) uses the classification of the linear algebraic subgroup of SL2(C) to obtain an algorithm that determines the Liouvillian solutions, which are the solutions that involve exponentials, indefinite integrals and solutions of polynomial equations. In particular, four cases happen: (1) H is conjugated to a subgroup of B = {( a b 0 a−1 ) , where a ∈ C∗, b ∈ C } , and there exists a Liouvillian solution of the form e ∫ X 0 , with f(X) ∈ C(X). (2) H is conjugated to a subgroup of D∞ = {( a 0 0 a−1 )⋃( 0 b−1 −b 0 ) , where a, b ∈ C∗ } , and there exists a Liouvillian solution of the form e ∫ X 0 , where f(X) is algebraic over C(X) of degree two and f(X) / ∈ C(X). (3) H is finite and all the solutions are algebraic over C(X). (4) H = SL2(C) and there are no Liouvillian solutions. Received by the editors October 3, 2011 and, in revised form, October 11, 2011 and April 12, 2012. 2010 Mathematics Subject Classification. Primary 34M15, 12H20, 34M03. Work partially supported by NFS CCF-0952591 and ANR-06-JCJC-0028. c ©2014 American Mathematical Society Reverts to public domain 28 years from publication