Application of the infinite matrix theory to the solvability of a system of differential equations

In this paper we deal with the solvability of the infinite system of differential equations x′(t) = ∆(λ)x(t) + b with x(0) = a, where ∆(λ) is the triangle defined by the infinite matrix whose the nonzero entries are [∆(λ)]nn = λn and [∆(λ)]n,n−1 = λn−1 for all n ∈ N, for a given sequence λ and a, b are two given infinite column matrices. We use a new method based on Laplace transformations to solve this system.