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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.

MSC:

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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