First order linear ordinary differential equations in associative algebras

Abstract

In this paper, we study the linear differential equation dx / dt = nΣ i=1 ai(t)xbi(t) + f(t) in an associative but non-commutative algebra A, where the bi(t) form a set of commuting A-valued functions expressed in a time-independent spectral basis consisting of mutually annihilating idempotents and nilpotents. Explicit new closed solutions are derived, and examples are presented to illustrate the theory.