Consider the regular Sturm-Liouville (S-L) system

$$(\mathcal{L} y)(x)-\lambda \omega(x) y(x)=0, \quad a \leqslant x \leqslant b$$

where

$$(\mathcal{L} y)(x):=-\left[p(x) y^{\prime}(x)\right]^{\prime}+q(x) y(x)$$

with $\omega(x)>0$ and $p(x)>0$ for all $x$ in $[a, b]$, and the boundary conditions on $y$ are

$$\left\{\begin{array}{l} A_{1} y(a)+A_{2} y^{\prime}(a)=0 \\ B_{1} y(b)+B_{2} y^{\prime}(b)=0 \end{array}\right.$$

Show that with these boundary conditions, $\mathcal{L}$ is self-adjoint. By considering $y \mathcal{L} y$, or otherwise, show that the eigenvalue $\lambda$ can be written as

$$\lambda=\frac{\int_{a}^{b}\left(p y^{\prime 2}+q y^{2}\right) d x-\left[p y y^{\prime}\right]_{a}^{b}}{\int_{a}^{b} \omega y^{2} d x}$$

Now suppose that $a=0$ and $b=\ell$, that $p(x)=1, q(x) \geqslant 0$ and $\omega(x)=1$ for all $x \in[0, \ell]$, and that $A_{1}=1, A_{2}=0, B_{1}=k \in \mathbb{R}^{+}$and $B_{2}=1$. Show that the eigenvalues of this regular S-L system are strictly positive. Assuming further that $q(x)=0$, solve the system explicitly, and with the aid of a graph, show that there exist infinitely many eigenvalues $\lambda_{1}<\lambda_{2}<\cdots<\lambda_{n}<\cdots$. Describe the behaviour of $\lambda_{n}$ as $n \rightarrow \infty$.