I really have no idea. I've never seen an equation like this.
Ok. This is a bit of a subtle point, and it is the essential idea behind separation of variables for pde's (we do not have a pde here, I know, but the essential idea is the same). The idea here is that each side of the equation has only one coordinate in it. Therefore, both sides must be constant. Otherwise, we could change just one side without changing the other, thus violating the equation. Does that make sense?
Yes it does. I'm familiar with the idea. But doesn't that work only if u is independent of v. But here that is not the case. If we set $\displaystyle u^{-1}(u)=t$
to be the inverse.
$\displaystyle v(t)=v(u^{-1}(u))=\alpha (u)$
Just like reparametrisation in differential geometry. You can't change the left hand size without changing the right hand size.
Ok. In that case, I'd agree that your original system of DE's is correct. I think you're also correct in thinking that the "separation of variables" trick I recommended won't work. Based on some preliminary research, I'd say that a complete analytic solution does not exist. You can do the first integral with the substitution, as outlined in your post #9. That reduced DE might be more stable with respect to numerical integration than the original DE.
I'm afraid that's as much as I know.
Cheers.