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.

Any conditions for $a,c,d$ ? From a numerical point of view, the problem is interesting. — Claude Leibovici, Aug 03 '18 at 15:12

score 1 · Answer 1 · answered Aug 03 '18 at 14:23

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.

answered Aug 03 '18 at 14:23

Claude Leibovici

260,315

This is awesome. Exactly what I was looking for. Thanks. – Bona Aug 05 '18 at 21:39

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

1 Answers1