I would like to know the general solution of $$x^2 y^{''}+x y^{'} -k^2 x^2 y=0\tag{1}$$ for $y(x)$.
(1) is related to the equation $$\quad x^2 y^{''}+x y^{'} - x^2 y=0$$ , which is the $n=0$ form of the modified Bessel equation
$$x^2 y^{''}+x y^{'} -( x^2+n^2) y=0$$
Other Information.
The material of this question turns up in fluid mechanics, in the theory of the instability of jets.
There is plenty of material on Wikipedia about the Bessel and related equations but I have not found there, the general solution I am looking for.