Euler's Equation

Euler's equation

[′ȯi·lərz i¦kwā·zhən] (mathematics) Euler-Lagrange equation

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.

单词	euler's equation
释义	Euler's Equation Euler's equation [′ȯi·lərz i¦kwā·zhən] (mathematics) Euler-Lagrange equation 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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