(i) The solution to the equation

$$\frac{d}{d x}\left(x \frac{d F}{d x}\right)+\alpha^{2} x F=0$$

that is regular at the origin is $F(x)=C J_{0}(\alpha x)$, where $\alpha$ is a real, positive parameter, $J_{0}$ is a Bessel function, and $C$ is an arbitrary constant. The Bessel function has infinitely many zeros: $J_{0}\left(\gamma_{k}\right)=0$ with $\gamma_{k}>0$, for $k=1,2, \ldots$. Show that

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

(where $\alpha$ and $\beta$ are real and positive) and deduce that

$$\int_{0}^{1} J_{0}\left(\gamma_{k} x\right) J_{0}\left(\gamma_{\ell} x\right) x d x=0, \quad k \neq \ell ; \quad \int_{0}^{1}\left(J_{0}\left(\gamma_{k} x\right)\right)^{2} x d x=\frac{1}{2}\left(J_{0}^{\prime}\left(\gamma_{k}\right)\right)^{2}$$

[Hint: For the second identity, consider $\alpha=\gamma_{k}$ and $\beta=\gamma_{k}+\epsilon$ with $\epsilon$ small.]

(ii) The displacement $z(r, t)$ of the membrane of a circular drum of unit radius obeys

$$\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial z}{\partial r}\right)=\frac{\partial^{2} z}{\partial t^{2}}, \quad z(1, t)=0$$

where $r$ is the radial coordinate on the membrane surface, $t$ is time (in certain units), and the displacement is assumed to have no angular dependence. At $t=0$ the drum is struck, so that

$$z(r, 0)=0, \quad \frac{\partial z}{\partial t}(r, 0)=\left\{\begin{array}{cc} U, & r<b \\ 0, & r>b \end{array}\right.$$

where $U$ and $b<1$ are constants. Show that the subsequent motion is given by

$$z(r, t)=\sum_{k=1}^{\infty} C_{k} J_{0}\left(\gamma_{k} r\right) \sin \left(\gamma_{k} t\right) \quad \text { where } \quad C_{k}=-2 b U \frac{J_{0}^{\prime}\left(\gamma_{k} b\right)}{\gamma_{k}^{2}\left(J_{0}^{\prime}\left(\gamma_{k}\right)\right)^{2}}$$