Definition:Euler's Equation for Vanishing Variation

Equation
Let $y(x)$ be a real function.

Let $F(x, y, z)$ be a real function belonging to $C^2$ w.r.t. all its variables.

Let $J[y]$ be a functional of the form

$\displaystyle \int_{a}^{b}F\left(x, y, y'\right) \mathrm{d}{x}$

Then Euler's Equation for Vanishing Variation is defined as

$\displaystyle F_y-\frac{\mathrm{d}{}}{\mathrm{d}{x}}F_{y'}=0 $