A novel *R-based perspective on solving ordinary differential equations

The real numbers, it is taught at universities, correspond to our idea of a continuum, although the hyperreal numbers are located in between'' the real numbers. The number $x + dx$, where $dx$ should be an infinitesimal number and $x$ real, is infinitesimally close to $x$ but infinitely’’ far away from all other real numbers. Analogously: If $f^{\mathrm{\prime }}(x_0)$ and $f(x_0)$ are given for a differentiable function $f:\mathbb{R}\to \mathbb{R}$ at $x_0\in \mathbb{R}$, we can not determine $f(x)$ at {\em any} point $x\in $\mathbb{R}$$differentfrom$x_0$. These points seem to be infinitely'' far away. That is one conceptual problem of solving differential equations in numerical mathematics. In this article, we will present a numerical algorithm to solve very simple initial value problems. However, the change of paradigm is, that we will not leave’’ the point $x_0$. Solving ordinary differential equations is like searching for recipes'' $f$. Instead of trying to find these recipes for values $x\in\mathbb{R}$, we will learn them from special relations in the monad’’ of $x_0$.