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

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

• Consider Taylor series expansion of $e^x$ say to 3 terms around zero, you could get a quadratic equation that may be useful as an approximate root. – NoChance Oct 4 '19 at 19:25
• Approximation is easy, though, I need the exact analytical solution :) – peterkey Oct 4 '19 at 19:29
• It helps to specify the domain of $x$, you could replace $e^x$ by a single fraction as in math.stackexchange.com/questions/71357/approximation-of-e-x but an closed form is probably not possible. – NoChance Oct 4 '19 at 19:33
• $$x\in [0;1]$$ – peterkey Oct 4 '19 at 19:42
• I assume that the approximation in the provided like is excellent. Plot it against the $e^x$ and see how good it is. – NoChance Oct 4 '19 at 19:48

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.