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

Given a basis of solutions to k ordinary linear differential equations ℓj[y]=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 ℓ[y]=0, where ℓ is the composite product ℓ=ℓ1ℓ2…ℓk. 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 ℓ1[y]=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.