Variation of parameters and solutions of composite products of linear differential equations

Given a basis of solutions to k ordinary linear differential equations ?jy]=0(j=1,2,…,k), we show how Green's functions can be used to construct a basis of solutions to the homogeneous differential equation ?y]=0, where ? is the composite product ?=?₁?₂…?k. The construction of these solutions is elementary and classical. In particular, we consider the special case when . Remarkably, in this case, if {y₁,y₂,…,yn} is a basis of ?₁y]=0, then our method produces a basis of for any k∈N. We illustrate our results with several classical differential equations and their special function solutions.