Graphic Solution of a Second-Order Differential Equation
This Demonstration shows the Euler–Cauchy method for approximating the solution of an initial value problem with a second-order differential equation. An example of such an equation is , with derivatives from now on always taken with respect to . This equation can be written as a pair of first-order equations, , .[more]
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.[less]
 V. I. Smirnoff, Lectures in Higher Mathematics (in Russian), vol. 2, Moscow: Nauka, 1967 p. 50.
 L. Euler, "De Integratione Aequationum Differentialium Per Approximationem," Institutionum Calculi Integralis Volumen Primum, 1768. www.math.dartmouth.edu/~euler/docs/originals/E342sec2ch7.pdf.