Algebrization of Nonautonomous Differential Equations

Abstract

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 $H:\mathrm{\Omega }\subset {\mathbb{R}}^2\times {\mathbb{R}}^2\to {\mathbb{R}}^2$ on an open set $\mathrm{\Omega }$ such that $F(t,w)=H(te,w)$ 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 $(\tau ,\xi )\in \mathrm{\Omega }$, 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 $(x(t),y(t))=\xi (te)$ of the planar system.