Electronic Journal of Differential Equations (Jan 2004)

First order linear ordinary differential equations in associative algebras

  • Gordon Erlebacher,
  • Garrret E. Sobczyk

Journal volume & issue
Vol. 2004, no. 01
pp. 1 – 18

Abstract

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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.

Keywords