On Solving Systems of Autonomous Ordinary Differential Equations by Reduction to a Variable of an Algebra

Abstract

A new technique for solving a certain class of systems of autonomous ordinary differential equations over 𝕂n is introduced (𝕂 being the real or complex field). The technique is based on two observations: (1), if 𝕂n has the structure of certain normed, associative, commutative, and with a unit, algebras 𝔸 over 𝕂, then there is a scheme for reducing the system of differential equations to an autonomous ordinary differential equation on one variable of the algebra; (2) a technique, previously introduced for solving differential equations over ℂ, is shown to work on the class mentioned in the previous paragraph. In particular it is shown that the algebras in question include algebras linearly equivalent to the tensor product of matrix algebras with certain normal forms.