Use the substitution $y=x^{p}$ to find the general solution of

$$\mathcal{L}_{x} y \equiv \frac{d^{2} y}{d x^{2}}-\frac{2}{x^{2}} y=0$$

Find the Green's function $G(x, \xi), 0<\xi<\infty$, which satisfies

$$\mathcal{L}_{x} G(x, \xi)=\delta(x-\xi)$$

for $x>0$, subject to the boundary conditions $G(x, \xi) \rightarrow 0$ as $x \rightarrow 0$ and as $x \rightarrow \infty$, for each fixed $\xi$.

Hence, find the solution of the equation

$$\mathcal{L}_{x} y= \begin{cases}1, & 0 \leqslant x<1, \\ 0, & x>1\end{cases}$$

subject to the same boundary conditions.

Verify that both forms of your solution satisfy the appropriate equation and boundary conditions, and match at $x=1$.