A paper I am working on starts with the choice of three parameters $n$, $l$, and $e$. These then determine three more parameters $r_a$, $r_b$, and $\epsilon$.
It would be more useful and meaningful if I could start with $r_a$, $r_b$, and $\epsilon$ and solve for $n$, $l$, and $e$. When the problem is simplified, its found that n has the following dependence:
$$ r_a^{(2-n)}+\epsilon r_a^2 = r_b^{(2-n)}+\epsilon r_b^2 $$
Is there anyway that one can solve for $n$? I have rewritten the equation in terms of the Lambert-W function, $W(z)e^{W(z)}=z$, however I'm still unable to find a closed form solution that can be written as a function $n=f(r_a,r_b,\epsilon)$, even one that includes the Lambert-W function. I feel like the W function was key, but that I am still missing something.
I guess my end-game is that if I can find such a function, any terms including W could be expanded using some iterative method for approximating W.
Additional thoughts: $r_b < r_a$ and both $r_{a,b} > 0$
