(a) Put the equation

$$x \frac{d^{2} u}{d x^{2}}+\frac{d u}{d x}+\lambda x u=0, \quad 0 \leqslant x \leqslant 1$$

into Sturm-Liouville form.

(b) Suppose $u_{n}(x)$ are eigenfunctions such that $u_{n}(x)$ are bounded as $x$ tends to zero and

$$x \frac{d^{2} u_{n}}{d x^{2}}+\frac{d u_{n}}{d x}+\lambda_{n} x u_{n}=0, \quad 0 \leqslant x \leqslant 1$$

Identify the weight function $w(x)$ and the most general boundary conditions on $u_{n}(x)$ which give the orthogonality relation

$$\left(\lambda_{m}-\lambda_{n}\right) \int_{0}^{1} u_{m}(x) w(x) u_{n}(x) d x=0$$

(c) The equation

$$x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}+x y=0, \quad x>0$$

has a solution $J_{0}(x)$ and a second solution which is not bounded at the origin. The zeros of $J_{0}(x)$ arranged in ascending order are $j_{n}, n=1,2, \ldots$. Given that $u_{n}(1)=0$, show that the eigenvalues of the Sturm-Liouville problem in (b) are $\lambda=j_{n}^{2}, n=1,2, \ldots$

(d) Using the differential equations for $J_{0}(\alpha x)$ and $J_{0}(\beta x)$ and integration by parts, show that

$$\int_{0}^{1} J_{0}(\alpha x) J_{0}(\beta x) x d x=\frac{\beta J_{0}(\alpha) J_{0}^{\prime}(\beta)-\alpha J_{0}(\beta) J_{0}^{\prime}(\alpha)}{\alpha^{2}-\beta^{2}} \quad(\alpha \neq \beta)$$