Starting from the Euler-Lagrange equation, show that a condition for

$$\int f\left(y, y^{\prime}\right) d x$$

to be stationary is

$$f-y^{\prime} \frac{\partial f}{\partial y^{\prime}}=\text { constant }$$

In the half-plane $y>0$, light has speed $c(y)=y+c_{0}$ where $c_{0}>0$. Find the equation for a light ray between $(-a, 0)$ and $(a, 0)$. Sketch the solution.