Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 01, pp. 1-18.

Title: First order linear ordinary differential equations in associative algebras

Authors: Gordon Erlebacher (Florida State Univ., Tallahassee, USA)
         Garrret E. Sobczyk (Univ. de las Americas, Mexico)

Abstract: 
  In this paper, we study the linear differential equation
  $$
  \frac{dx}{dt}=\sum_{i=1}^n a_i(t) x b_i(t) + f(t)
  $$
  in an associative but non-commutative algebra $\mathcal{A}$,
  where the $b_i(t)$ form a set of commuting $\mathcal{A}$-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.

Submitted September 6, 2003. Published January 2, 2004.
Math Subject Classifications: 15A33, 15A66, 34G10, 39B12.
Key Words: Associative algebra;  factor ring; idempotent;
   differential equation; nilpotent; spectral basis; Toeplitz matrix.