Could someone please help me with the following?
I have a linear operator
$Ly=-x^{-2}(x^2y')'+y$, where $" ' "$ denotes $d\over dx$
I need to find the solution $y(x)$ to the forced equation $Ly=F(x)$
subject to boundary conditions $y(x)$ is bounded as $x\to 0, \infty$
where $$F(x) = \begin{cases}1 & x\in[0,x_0]\\0 &x>x_0\end{cases}$$
The solution, I was told, should have the form:
$$y(x) =\begin{cases} {\alpha\sinh x\over x}+1 &x\in [0, x_0]\\ {\beta e^{-x}\over x} &x>x_0 \end{cases}$$
Thanks in advance for any help!