(i) Briefly describe the Sturm-Liouville form of an eigenfunction equation for real valued functions with a linear, second-order ordinary differential operator. Briefly summarize the properties of the solutions.

(ii) Derive the condition for self-adjointness of the differential operator in (i) in terms of the boundary conditions of solutions $y_{1}, y_{2}$ to the Sturm-Liouville equation. Give at least three types of boundary conditions for which the condition for self-adjointness is satisfied.

(iii) Consider the inhomogeneous Sturm-Liouville equation with weighted linear term

$$\frac{1}{w(x)} \frac{d}{d x}\left(p(x) \frac{d y}{d x}\right)-\frac{q(x)}{w(x)} y-\lambda y=f(x),$$

on the interval $a \leqslant x \leqslant b$, where $p$ and $q$ are real functions on $[a, b]$ and $w$ is the weighting function. Let $G(x, \xi)$ be a Green's function satisfying

$$\frac{d}{d x}\left(p(x) \frac{d G}{d x}\right)-q(x) G(x, \xi)=\delta(x-\xi)$$

Let solutions $y$ and the Green's function $G$ satisfy the same boundary conditions of the form $\alpha y^{\prime}+\beta y=0$ at $x=a, \mu y^{\prime}+\nu y=0$ at $x=b(\alpha, \beta$ are not both zero and $\mu, \nu$ are not both zero) and likewise for $G$ for the same constants $\alpha, \beta, \mu$ and $\nu$. Show that the Sturm-Liouville equation can be written as a so-called Fredholm integral equation of the form

$$\psi(\xi)=U(\xi)+\lambda \int_{a}^{b} K(x, \xi) \psi(x) d x$$

where $K(x, \xi)=\sqrt{w(\xi) w(x)} G(x, \xi), \psi=\sqrt{w} y$ and $U$ depends on $K, w$ and the forcing term $f$. Write down $U$ in terms of an integral involving $f, K$ and $w$.

(iv) Derive the Fredholm integral equation for the Sturm-Liouville equation on the interval $[0,1]$

$$\frac{d^{2} y}{d x^{2}}-\lambda y=0,$$

with $y(0)=y(1)=0$.