Euler's equation[′ȯi·lərz i¦kwā·zhən]
(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.