# Euler's Equation

## Euler's equation

[′ȯi·lərz i¦kwā·zhən]## Euler’s Equation

(**1**) A differential equation of the form

where *a*_{0}, . . ., *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*) = *a*_{0}*x*^{4} + *a*_{1}*x*^{3} + *a*_{2}*x*^{2} + *a*_{3}*x* + *a*_{4} and *Y*(*y*) = *a*_{0}*y*^{4} + *a*_{1}*y*^{3} + *a*_{2}*y*^{2} + *a*_{3}*y* + *a*_{4}. 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.