Hi, I have a differential equation that I'm trying to solve my substitution. I've got most of it, but I can figure out how to end it properly.

The equation: x y' - y = y [1 - ln^{2}(y/x)]

I divide by x

let v = y/x

y' = v + xv'

sub v and y' in the equation to get:

dx/x = dv/[v(1-ln^{2}v)]

let u = ln v

du = dv/v

sub these to get:

dx/x = du/(1-u^{2})

Integrating this I get:

ln |x| = 1/2 * ln | (1+u)/(1-u) |

reverse substituting v:

ln |x| = 1/2 * ln | (1+ ln v) / (1 - ln v ) |

reverse substituting y:

ln |x| = 1/2 * ln | [1+ ln (y/x)] / [1- ln (y/x)] |

How do I proceed after this. Is there any way to get the solution in terms of y?

Or should I use a different approach from the beginning itself?

Any help is appreciated, thanks