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,ζ) and h2 (ζ)=h(τ,ζ) are lorch differentiable with respect to a for all (τ,)ε ω,where τ and represent variables in a. under these conditions the solutions (τ) of the differential equation d/dτ=h(τ,ζ) over a define solutions (x(t),y(t))=(te) of the planar system. © 2015 maría aracelia alcorta-garcía et al.