Any help how this equation ( Eq. B3 in paper ) :

$$ \phi’’(\eta) + \frac{6(1+\omega)}{1+3\omega)} \frac{1}{\eta} \phi’(\eta) + \omega k^2 \phi(\eta) =0 $$

Has been solved by Bessel’s function with a solution:

$ \phi(\eta) = y^{-\alpha} [ C_1 J_\alpha (y) + C_2 Y_\alpha (y) ], ~~~ y = \sqrt{\omega} k \eta,~~~~~ \alpha= \frac{1}{2} \left( \frac{5+3\omega}{1+3\omega} \right). $

I spent much time reading about Bessel’s function , but there are different cases. Here $\omega$ is a constant and $k$ is the wave number.