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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.

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  • $\begingroup$ 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. $\endgroup$
    – Simply Beautiful Art
    Jun 4, 2017 at 18:32
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    $\begingroup$ 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. $\endgroup$
    – Donald Splutterwit
    Jun 4, 2017 at 18:37
  • $\begingroup$ 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. $\endgroup$
    – Noomkwah
    Jun 4, 2017 at 19:08
  • $\begingroup$ For $(n\notin\{0,1\})\land(c\neq 0)$, neither $(x+a)n+e^x=c$ nor $x^n+e^x=c$ are solvable with only elementary functions and Lambert W. $\endgroup$
    – IV_
    Aug 4, 2022 at 14:42

1 Answer 1

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For applying only Lambert W and elementary functions, your equation should be transformable to 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 partial 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.

For $((a\neq 0)\lor(b\neq 0))\land(c\neq 0)$, your equation cannot be solved in terms of elementary functions and/or Lambert W. But the equation is solvable by generalized Lambert W:

$$x=W\left(^{-a\ \ -b}_{};c\right)$$

$$x=W^{(p)}\left(^{-a\ \ -b}_{+1\ \ +1};c\right)$$

[Mezö 2017] Mezö, I.: On the structure of the solution set of a generalized Euler-Lambert equation. J. Math. Anal. Appl. 455 (2017) (1) 538-553

[Mezö/Baricz 2017] Mezö, I.; Baricz, A.: On the generalization of the Lambert W function. Transact. Amer. Math. Soc. 369 (2017) (11) 7917–7934 (On the generalization of the Lambert W function with applications in theoretical physics. 2015)

[Castle 2018] Castle, P.: Taylor series for generalized Lambert W functions. 2018

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