How do I solve $(a+x)e^{-bx} = c$?

Question

How do I solve $$(a+x)\mathrm e^{-bx} = c?$$ I tried looking into Lambert W function but was not able to solve it.

Answer 1

It's not difficult to express the solution with the W function

$$(a+x)e^{-bx} = c \Leftrightarrow (-ba-bx)e^{-bx} = -bc$$

$$ \Leftrightarrow (-ba-bx)e^{-ba-bx} = -bce^{-ba}$$

So we have $-ba-bx = W( -bce^{-ba} )$

Hence

$$x = -a-\frac{W( -bce^{-ba} )}{b}$$