First order linear ordinary differential equations in associative algebras

Abstract

<p>In this paper, we study the linear differential equation</p> <pre>dx/ dt = Σⁿ_i=1 a_i(t)xb_i(t) + ƒ(t)</pre> <p>in an associative but non-commutative algebra A, where the b_i(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.</p>