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 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 pairs are the coordinates of points , , …, that form the so-called Euler's polygonal line that approximates the graph of the function . In the same way, the pairs are the coordinates of points , , …, that form Euler's polygonal line, which approximates the graph of the function .
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.

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References
[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. www.math.dartmouth.edu/~euler/docs/originals/E342sec2ch7.pdf.
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