$exp(x)-\frac{x^{2}}{a}+\frac{x}{b}-1=0$

Question

I have an equation that can be very likely solved with the Lambert function but looks a bit messy:

$exp(x)-\frac{x^2}{a}+\frac{x}{b}-1=0$, and $a>0$, $b\geq1$ (if constraints help).

Any idea how to get x out of this in the closed form?

Cheers, p

Answer 1

Unless factorization can be done, this requires a generalization of the Lambert W function. See Taylor series for generalized Lambert W function. In essence you can get a series expansion for $x$ in terms of the other variables using Lagrange inversion theorem, but it is likely messy.

Aside from that, not much more can be done to the problem other than numerical computation.