Algebrization of nonautonomous differential equations. (English) Zbl 1382.12002

Summary: Given a planar system of nonautonomous ordinary differential equations, \(dw/dt = F(t, w)\), conditions are given for the existence of an associative commutative unital algebra \(\mathbb{A}\) with unit \(e\) and a function \(H: \Omega \subset \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}^2\) on an open set \(\Omega\) such that \(F(t, w) = H(t e, w)\) and the maps \(H_1(\tau) = H(\tau, \xi)\) and \(H_2(\xi) = H(\tau, \xi)\) are Lorch differentiable with respect to \(\mathbb{A}\) for all \((\tau, \xi) \in \Omega\), where \(\tau\) and \(\xi\) represent variables in \(\mathbb{A}\). Under these conditions the solutions \(\xi(\tau)\) of the differential equation \(d \xi / d \tau = H(\tau, \xi)\) over \(\mathbb{A}\) define solutions \((x(t), y(t)) = \xi(t e)\) of the planar system.


12H20 Abstract differential equations
34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain
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