Euler's Equation

Euler's equation

[′ȯi·lərz i¦kwā·zhən]

(mathematics)

Euler’s Equation

(1) A differential equation of the form

where a₀, . . ., a_n are constants. When x > 0, equation (*) reduces, by the substitution x = e^t, 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.

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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.

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The latter requires application of Euler's equation from the calculus of variations.

In order to make the problem and its difficulties in application clearer we write Euler's equation, the necessary condition for the maximization of an integral over a given finite time span.

United States and German real capital formation and social investment in the sciences and humanities

b] obtained by Southwell's method and Engesser's tangent modulus are close to that predicted by Euler's equation.

Bending properties of medium-density fiberboard and plywood obtained by compression bending test

The simplest estimate, using Euler's equation for the buckling force on a beam (8), the bending stiffness of a single microtubule decorated with associated proteins [~5 x [10.

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He converted the problem into an equivalent mathematical form that enabled him to pick out candidates likely to satisfy Euler's equation.

No one knows whether another set of numbers, somewhere between those found by Frye and those discovered by Elkies, fits Euler's equation.

The curious power of large numbers

Here, the authors complete the proof of the envelope theorem for the class of variational problems concerned, using the corresponding Euler's equation.

Completing the "one line proof" of the dynamic envelope theorem

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.

Real Analysis and Applications - Including Fourier Series and the Calculus of Variations

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.

Wave forces on offshore structures

Week 6: Euler's equations, the rotator, the symmetric free top, Feynman's plate experiment, geometry of three-dimensional reconstruction of plate from image

Methods of applied mathematics: honors mathematics 450 and 451 each 3 credit hours

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.