More generally, the method to be described works for any system of two first-order differential equations , with initial conditions , . The particular kinds of systems used as examples here, , reduce to that general type by introducing to get the system , .

The method consists of simultaneously calculating approximations of (cyan) and (green):

,

, .

The Euler method is the most basic approximation method. The Demonstration compares it with more advanced methods given by the built-in Mathematica function NDSolve.