# Equations solvable by Lambert Function

I have learned that the Lambert function can be used to solve some equations. These equations are equivalent to : \begin{equation} (x+a)(x+b)e^x = c \end{equation} Is this correct ? In that case, how can we solve these equations ?

Thanks for answers.

• No, most of the time, it's not possible to solve a quadratic-exponential mix unless you have a perfect square e.g. $a=b$. The Lambert W function can only solve the general linear-exponential problems. – Simply Beautiful Art Jun 4 '17 at 18:32
• If you can rearrange to $f(x) e^{f(x)} = z$ then you can solve using the Lambert function ... I think you will struggle to do this with your equation. – Donald Splutterwit Jun 4 '17 at 18:37
• In that case, is it possible to solve this equation ?\begin{equation}(x+a)^n + e^x = c\end{equation} or \begin{equation}x^n + e^x = c\end{equation} if the first equation is not solvable. – Keryann Massin Jun 4 '17 at 19:08
• For $n\notin\{0,1\}$ with $n\in\mathbb{Z}$, neither $(x+a)^{n}+e^{x}=c$ nor $x^{n}+e^{x}=c$ are solvable with only elementary functions or with only elementary functions and LambertW. This equations cannot be brought to the form of equation (1) of my answer. – IV_ Nov 20 '17 at 17:27

## 1 Answer

For applying only Lambert W and elementary functions, your equation should be in the form

$$f_1(f_2(x)e^{f_2(x)})=c,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$

where $c$ constant and $f_1$ and $f_2$ are elementary functions with a suitable elementary local inverse. The problem of existence of elementary inverses of elementary functions is solved in Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90.

If your equation cannot be brought to the form of equation (1), you need a predefined generalization of Lambert W.

Maybe the following helps:

Wikipedia: Lambert W function - Generalizations

Mező, I.; Baricz, Á.: On the generalization of the Lambert W function. Trans. Amer. Math. Soc. 369 (2017) 7917-7934