On the Constructive Inverse Problem in Differential Galois Theory
  We give sufficient conditions for a linear differential equation to have a given semisimple group as its Galois group. For any linear algebraic group G given as a semidirect product of a finite subgroup and a normal subgroup that is a product of groups of type An, Cn, Dn, E6, or E7, we construct a differential equation over C(x) having Galois group G. 
William J. Cook, Claude Mitschi, Michael F. Singer