(a) Consider the general self-adjoint problem for $y(x)$ on $[a, b]$ :

$$-\frac{d}{d x}\left[p(x) \frac{d}{d x} y\right]+q(x) y=\lambda w(x) y ; \quad y(a)=y(b)=0$$

where $\lambda$ is the eigenvalue, and $w(x)>0$. Prove that eigenfunctions associated with distinct eigenvalues are orthogonal with respect to a particular inner product which you should define carefully.

(b) Consider the problem for $y(x)$ given by

$$x y^{\prime \prime}+3 y^{\prime}+\left(\frac{1+\lambda}{x}\right) y=0 ; \quad y(1)=y(e)=0 .$$

(i) Recast this problem into self-adjoint form.

(ii) Calculate the complete set of eigenfunctions and associated eigenvalues for this problem. [Hint: You may find it useful to make the substitution $\left.x=e^{s} .\right]$

(iii) Verify that the eigenfunctions associated with distinct eigenvalues are indeed orthogonal.