Stability Analysis of Nonlinear Bessel-type ODE

[player100]

Is there an approach to the following 2nd order nonlinear ODE?
$$xy'' + 2 y' = y^2 - k^2$$
I am interested in learning how to analyze for asymptotic behavior, proof of existence, etc.

 

[player100]

I can transform this equation into a more standard form: $$u^2 z_{uu} - u z_u + u^2 z = z^2.$$ I found a paper that kind of answers my question: http://www.sciencedirect.com/science/article/pii/0898122196000569

It would be nice to show that the solution is Lyapunov stable about z(u=0) = 0... Anyone know how to do that?