Starting from the Euler-Lagrange equations, show that the condition for the variation of the integral $\int I\left(y, y^{\prime}\right) \mathrm{d} x$ to be stationary is

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

In a medium with speed of light $c(y)$ the ray path taken by a light signal between two points satisfies the condition that the time taken is stationary. Consider the region $0<y<\infty$ and suppose $c(y)=e^{\lambda y}$. Derive the equation for the light ray path $y(x)$. Obtain the solution of this equation and show that the light ray between $(-a, 0)$ and $(a, 0)$ is given by

$$e^{\lambda y}=\frac{\cos \lambda x}{\cos \lambda a},$$

if $\lambda a<\frac{\pi}{2}$.

Sketch the path for $\lambda a$ close to $\frac{\pi}{2}$ and evaluate the time taken for a light signal between these points.

[The substitution $u=k e^{\lambda y}$, for some constant $k$, should prove useful in solving the differential equation.]