Obtain the Green function $G(x, \xi)$ satisfying

$$G^{\prime \prime}+\frac{2}{x} G^{\prime}+k^{2} G=\delta(x-\xi),$$

where $k$ is real, subject to the boundary conditions

$$\begin{array}{rll} G \text { is finite } & \text { at } & x=0, \\ G=0 & \text { at } & x=1 . \end{array}$$

[Hint: You may find the substitution $G=H / x$ helpful.]

Use the Green function to determine that the solution of the differential equation

$$y^{\prime \prime}+\frac{2}{x} y^{\prime}+k^{2} y=1,$$

subject to the boundary conditions

$$\begin{array}{rll} y \text { is finite } & \text { at } & x=0, \\ y=0 & \text { at } & x=1, \end{array}$$

is

$$y=\frac{1}{k^{2}}\left[1-\frac{\sin k x}{x \sin k}\right]$$