Approximation of exponential expression

Question

I have given the equation

$$au_x-ap+1+e^{-ap+b}=0,$$

where $p>0$ is the unknown. $u_x$ denotes the derivative of a given function, $a$ and $b$ are merely constants. I want to express $p$ explicitly using some sort of good approximation, such that in the end I get something like $$p(u_x)=...$$

Any chance to find something like that?

Answer 1

Expand the $p$ terms using the Taylor approximation, then use the Lagrange inversion formula https://en.wikipedia.org/wiki/Lagrange_inversion_theorem.

https://mathematica.stackexchange.com/q/166103/58644

Answer 2

Well, you have $$ -\exp(b-au_x-1) = (au_x-ap+1)\exp(ap-au_x-1) $$ so you can "invert" this using the Lambert-W function $$ au_x-ap+1 = W(-\exp(b-au_x-1)) $$ and hence express $p=p(a_u)$.