Quoting from Gelfand page 40:
Hence, the equations (15) cannot be independent, and in fact it is easily verified that they are connected by the identity $\dot{x}\left(\Phi_x - \frac{d}{dt}\Phi_{\dot{x}}\right) + \dot{y}\left(\Phi_y - \frac{d}{dt}\Phi_{\dot{y}}\right) = 0$
I wonder if someone could give a derivation for this identity?
For more information and clarity, we have that $\Phi(x,y,\dot x,\dot y) := F(x,y,\frac{\dot y}{\dot x})\dot x$, and for the functional $\int_{t_0}^{t_1}F(x,y,\frac{\dot y}{\dot x})\dot xdx = \int^{t_1}_{t_0}\Phi(x,y,\dot x,\dot y) dt$ to achieve an extremum, the condition $F_y - \frac{d}{dx}F{_y'} = 0$ should be equivalent to equation (15) $\Phi_x - \frac{d}{dt}\Phi_{\dot{x}} = 0, \Phi_y - \frac{d}{dt}\Phi_{\dot{y}} = 0$. However, I wonder how to show this through the identity provided aboved, which is claimed to be true in the book.