I have to solve Euler-Lagrange for $\displaystyle I[y] = \int \frac{1}{c_{0}} (1 - ky)^{ \frac{1}{2}} (1 + y' ^{2}) ^{ \frac{1}{2}} \,dx $.

This is for a problem of a medium through which light passes in 0 < y < h and we have that 0 < k < (1/h).

So we could do standard Euler-Lagrange and get $\displaystyle y'' (1 - ky) + \frac{k}{2} (1 + y' ^{2}) = 0 $, however I don't see how to solve this. If we note that there is no x-dependence then we can use that $\displaystyle y' f_{y'} - f = C $ and get that $\displaystyle - \sqrt{ \frac{ (1 - ky) }{ ( 1 + y' ^{2} ) } } = Cc_{0} $ however this does not seem good either. I am looking to show that the solution, y(x) is a parabola.