Euler's Equation

Euler's equation

[′ȯi·lərz i¦kwā·zhən]

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.

References in periodicals archive ?
Many of these entities have been given simple and ambiguous names such as Euler's function, Euler's equation, and Euler's formula.
b] obtained by Southwell's method and Engesser's tangent modulus are close to that predicted by Euler's equation.
He converted the problem into an equivalent mathematical form that enabled him to pick out candidates likely to satisfy Euler's equation.
He then moves to applications in Fourier series, including strings and springs, and devotes the rest of the text to the calculus of variations, including Euler's Equation, harmonic functions, Hamilton's Action and Lagranges' Equation, finishing with nonEuclidean geometry and some fun with relativity.
The classes of offshore structures are described in a short introduction, followed by a review of fundamental equations and concepts of motion, rotational and irrotational flows, velocity potential, Euler's equations, steam function and other concepts.
After presenting the foundations in a form free from any coordinate system, the author follows with a chapter on the technique of writing Navier-Stokes equations and Euler's equations in general steady and nonsteady curvilinear coordinates, as well as the essential aspects of vorticity and stream functions.