How do I solve $$(a+x)\mathrm e^{-bx} = c?$$ I tried looking into Lambert W function but was not able to solve it.
How do I solve $(a+x)e^{-bx} = c$?
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exponential-function
lambert-w
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1Note that $z=-b(x+a)$ solves $ze^z=-bce^{-ba}$ and apply the definition of Lambert W (for no gain except a useless pseudo-explicit formula...). – 2017-02-17
1 Answers
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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
$$x = -a-\frac{W( -bce^{-ba} )}{b}$$