First order linear ordinary differential equations in associative algebras. (English) Zbl 1042.34001

Summary: Here, 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 noncommutative 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.


34A05 Explicit solutions, first integrals of ordinary differential equations
15B33 Matrices over special rings (quaternions, finite fields, etc.)
15A66 Clifford algebras, spinors
34A99 General theory for ordinary differential equations
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