I am looking after the solution of the following differential equation:
$ \partial_y \partial_y \psi + \frac{\partial_y \psi}{y} + \left( \frac{\omega^2 }{y^2} - \frac{\Lambda}{y} \right) \psi = 0 $
where both $\omega$ and $\Lambda$ are constants.
I know *cough, Mathematica, cough * that the solution to this should be given in terms of (Modified) Bessel functions. However I am having a really hard time trying to show this.
As usual, I tried a variable change with a free parameter $P$ as:
$\psi(y) = \Psi(x)$ with $x=y^P$
where
$\frac{d \psi}{dy} = \frac{d \Psi(x)}{dx} \frac{dx}{dy} = \Psi' P y^{P-1}$
to get
$\Psi'' + y^{-P} \Psi' + \left( \frac{\omega^2}{P^2} y^{-2P} - \frac{\Lambda}{P^2} y^{-2P+1} \right) \Psi=0$
where, usually, we can choose a suitable $P$ to make this equation resemble a special function. However I cannot see anything from here.
How do I solve it, or make it resemble something Bessel-like?