How to solve following equation $a(x-2)e^x-c x-d=0$ (Maybe using Lambert W function)

Question

How to solve following equation $a(x-2)e^x-c x-d=0$. I know that the equation like $a(x-2)e^x-d=0$ can solve using Lambert W function, but with this equation I'm confused, and I can solve.

Answer 1

Assuming $c \neq 0$, write your equation as $$e^{-x}=\frac a c\, \frac{x-2}{ x+\frac dc}$$ and have a look at equation $(4)$ in this paper about the generalization of the Lambert $W$ function.

From a practical point of view, consider numerical methods.