Write down the Euler-Lagrange equation for the variational problem for $y(x)$ that extremizes the integral $I$ defined as

$$I=\int_{x_{1}}^{x_{2}} f\left(x, y, y^{\prime}\right) d x$$

with boundary conditions $y\left(x_{1}\right)=y_{1}, y\left(x_{2}\right)=y_{2}$, where $y_{1}$ and $y_{2}$ are positive constants such that $y_{2}>y_{1}$, with $x_{2}>x_{1}$. Find a first integral of the equation when $f$ is independent of $y$, i.e. $f=f\left(x, y^{\prime}\right)$.

A light ray moves in the $(x, y)$ plane from $\left(x_{1}, y_{1}\right)$ to $\left(x_{2}, y_{2}\right)$ with speed $c(x)$ taking a time $T$. Show that the equation of the path that makes $T$ an extremum satisfies

$$\frac{d y}{d x}=\frac{c(x)}{\sqrt{k^{2}-c^{2}(x)}}$$

where $k$ is a constant and write down an integral relating $k, x_{1}, x_{2}, y_{1}$ and $y_{2}$.

When $c(x)=a x$ where $a$ is a constant and $k=a x_{2}$, show that the path is given by

$$\left(y_{2}-y\right)^{2}=x_{2}^{2}-x^{2} .$$