(a) Consider the functional

$$I[y]=\int_{a}^{b} L\left(y, y^{\prime} ; x\right) d x$$

where $0<a<b$, and $y(x)$ is subject to the requirement that $y(a)$ and $y(b)$ are some fixed constants. Derive the equation satisfied by $y(x)$ when $\delta I=0$ for all variations $\delta y$ that respect the boundary conditions.

(b) Consider the function

$$L\left(y, y^{\prime} ; x\right)=\frac{\sqrt{1+y^{\prime 2}}}{x} .$$

Verify that, if $y(x)$ describes an arc of a circle, with centre on the $y$-axis, then $\delta I=0$.

(c) Consider the function

$$L\left(y, y^{\prime} ; x\right)=\frac{\sqrt{1+y^{\prime 2}}}{y}$$

Find $y(x)$ such that $\delta I=0$ subject to the requirement that $y(a)=a$ and $y(b)=\sqrt{2 a b-b^{2}}$, with $b<2 a$. Sketch the curve $y(x)$.