The question is. Determine the solutions of the Euler-Lagrange equation when
$$f(x,y,y')=y$$
Which is fairly straightforward. At this point i would note the Beltrami equation is a simplification of the EL eqn.
Via EL $$\frac{\partial f}{\partial y}-\frac{d}{dx}[\frac{\partial f}{\partial y'}] = 1$$ This implies $1 = 0$ and so we have no solutions.
But via the Beltrami equation we have $$f-\frac{\partial f}{\partial y'}y'=y=0$$ which does have a solution (unless I'm being thick in which case please say so)
Any explanation would be great. (Side note i apologise for formatting. I'm using mobile app and mathjax doesn't always render on preview for me).