formula

$\frac{\partial F(y,y')}{\partial y}-\frac{d}{dx}(\frac{\partial F(y,y')}{\partial y'})=0\\ \frac{1}{y'}\frac{dF}{dx}-\frac{1}{y'}\frac{\partial F}{\partial y'}\frac{dy'}{dx}-\frac{d}{dx}(\frac{\partial F}{\partial y'})=0\\ \frac{1}{dx}(F-y'\frac{\partial F}{\partial y'})=0\\ \frac{1}{\sqrt{2gy(1+y'^2)}}=C(const.)\\ y(1+y'^2)=\frac{1}{2gC^2}$