Algebrization of Nonautonomous Differential Equations

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 A with unit e and a function H:Ω⊂R2×R2→R2 on an open set Ω such that F(t, w) = H(te, w) and the maps H1(τ) = H(τ, ξ) and H2(ξ) = H(τ, ξ) are Lorch differentiable with respect to A for all (τ, ξ) ∈ Ω, where τ and ξ represent variables in A. Under these conditions the solutions ξ(τ) of the differential equation dξ/dτ = H(τ, ξ) over A define solutions (x(t), y(t)) = ξ(te) of the planar system.