Modified Bessel Differential Equation

The second-order ordinary differential equation

$$x^2 {d^2y\over dx^2}+x {dy\over dx}-(x^2+n^2)y = 0.$$

The solutions are the Modified Bessel Functions of the First and Second Kinds. If $n=0$, the modified Bessel differential equation becomes

$$x^2 {d^2y \over dx^2}+x {dy\over dx}-x^2 y = 0,$$

which can also be written

$${d\over dx}\left({x{dy\over dx}}\right)=xy.$$