I am trying to solve the following equation ($ellin mathbb{N}$)

$$f”(x)+left(1-dfrac{ell(ell+1)}{x^2}+frac{2}{x}right)f(x)=0,$$

with these two asymptotic boundary conditions (b.c.) on the function and its derivative:

$$lim_{xrightarrow0}f'(x)=(ell+1)(f(x)/x),$$

$$lim_{xrightarrowinfty}f(x)=frac{1}{2i}left(e^{ix}-e^{-i(x-ellpi)}right).$$

The equation

`DiffEq = f''(x) + (1 - l(l+1)/x^2 + 2/x) f(x) == 0`

can be analytically solved with `DSolve`

, but I do not understand how to impose these two “analytical” b.c. to get rid of the two integration constants. I mean, they are not merely “numerical” b.c. and therefore I cannot simply do something like `DSolve({DiffEq,f(a)==b,f(c)==d},f(x),x)`

, with `a,b,c,d`

being some numbers.

Does anybody have an idea?

Thanks a lot!