Ah yes, there's this method of $N(x,y)dx+M(x,y)dy=dF$ which is learnt in undergraduate, but I can't remember how this method conditues.

I am sure you can find some special function solution, if it has been studied before. Perhaps looking at something in modular forms and functions, sometimes these sort of ODEs pop up there like Weirestrass P-function.

@GHfromMO yes it is Sidon. I remember quite interesting problems.

Try math.stackexchange.com

This is why it's called "Lie". :-) (in case you don't understand I made a joke).

I don't understand why would one need $\ne$ sign, when you have both signs of $=$ and $\neg$ in there?

A lot of good stuff to read, thanks!

@SamZbarsky let's assume something like: $y(-\alpha)=0$ and $y(\alpha)=1$ for all the iterations.

by all means let get ugly... :-D

I guess you can do repeated limits, as in $\lim_{(x,t)\rightarrow (0,\infty)}u(x,t)/\sqrt{t}\log t$, and then if you knew that at $t\to \infty$ $u(x,t)$ approaches $\pm \infty$ then you could have used the PDE with L'hopital; but this is quite a messy computation.