# A Nonlinear Bessel-type ODE

1. Jul 12, 2016

### 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.

2. Jul 18, 2016

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Jul 19, 2016

### 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?