Graphic Solution of a Second-Order Differential Equation

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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 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.


Contributed by: Izidor Hafner (January 2014)
Open content licensed under CC BY-NC-SA




[1] V. I. Smirnoff, Lectures in Higher Mathematics (in Russian), vol. 2, Moscow: Nauka, 1967 p. 50.

[2] L. Euler, "De Integratione Aequationum Differentialium Per Approximationem," Institutionum Calculi Integralis Volumen Primum, 1768.

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