Euler's Equation

Euler's equation

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McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Euler’s Equation


(1) A differential equation of the form

where a0, . . ., an are constants. When x > 0, equation (*) reduces, by the substitution x = et, to a linear differential equation with constant coefficients. Equation (*) was studied by L. Euler in 1740. The equation

reduces to it through the substitution x’ = ax + b.

(2) A differential equation of the form

where X(x) = a0x4 + a1x3 + a2x2 + a3x + a4 and Y(y) = a0y4 + a1y3 + a2y2 + a3y + a4. Euler considered this equation in a number of works, the first of which dates from 1753. He showed that the general solution of the equation is a symmetric polynomial of degree 4 in x and y. This result provided the basis for the theory of elliptic integrals.

(3) A differential equation of the form

which is used in the calculus of variations to search for extrema of the integral

The equation was derived by L. Euler in 1744.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
Popa, "Hyers-Ulam stability of Euler's equation," Applied Mathematics Letters, vol.
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