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I am solving this equation $$(ax+b)\exp(-cx) = (fx+d)$$ using generalized Lambert W function and $a,b,c,d,f,d$ are all real-valued.

I drew this equation's graph using Matlab, and I confirmed this equation passes $0$. But when I solve it using generalized Lambert W function, it doesn't come out with an accurate value. And I know Generalized Lambert W function solution sometimes has complex numbers. But I want to obtain only real number.

In this situation, I want to know what should I check.

IV_
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SEUNGMIN SIM
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1 Answers1

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$$(ax+b)e^{-cx}=(fx+d)$$

We see, this equation is a polynomial equation of more than one algebraically independent monomials ($x,e^{cx}$) and with no univariate factor. We therefore don't know how to rearrange the equation for $x$ by applying only finite numbers of elementary functions (operations) we can read from the equation.

If the equation is irreducible for algebraic $a,b,c,d,f$, the equation cannot have solutions that are elementary numbers.

$$\frac{ax+b}{fx+d}e^{-cx}=1$$

We see, in the general case, we cannot solve this equation in terms of Lambert W, but in terms of Generalized Lambert W.

a)

$x\to -\frac{t}{c}$: $$\frac{-\frac{a}{c}t+b}{-\frac{f}{c}t+d}e^t=1$$ $$\frac{-\frac{a}{c}(t-\frac{bc}{a})}{-\frac{f}{c}(t-\frac{cd}{f})}e^t=1$$ $$\frac{t-\frac{bc}{a}}{t-\frac{cd}{f}}e^t=\frac{f}{a}$$ $$t=W\left(^\frac{bc}{a}_\frac{cd}{f};\frac{f}{a}\right)$$ $$x=-\frac{1}{c}W\left(^\frac{bc}{a}_\frac{cd}{f};\frac{f}{a}\right)$$

b)

according to theorem 3 of [Mező/Baricz 2017]:

$W_r$ is the $r$-Lambert function.

$$x=-\frac{b}{a}-\frac{1}{c}W_{-ae^{-\frac{bc}{a}}}\left(-ace^{-\frac{bc}{a}}\left(-\frac{b}{a}+\frac{d}{f}\right)\right)$$ $\ $

So we have a closed form for $x$, and the series representations of Generalized Lambert W give some hints for calculating $x$.
see e.g. [Mező/Baricz 2017] section 3: "The case of one upper and one lower parameter" and theorem 3.

see also: Interpreting/understanding the lambertW on Maple software

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

[Mező/Baricz 2017] Mező, I.; Baricz, Á.: 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

IV_
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  • Thank you so much. There was a mistake in substitution. But when I input this solution to Matlab, the value is not real-valued, complex-valued. I think, because of Lambert W function. $W_k$, k is not -1 or 0. When k is not -1 or 0, how can I approximate this to real-valued? – SEUNGMIN SIM Oct 03 '23 at 16:23
  • I am sorry, your solution is correct but I made a mistake during the substitution. I know this solution is represented to generalized Lambert W function. but generalied lambert W function solution sometime has complex number. But I want to real number. So if any way to solve this, could you tell me? – SEUNGMIN SIM Oct 03 '23 at 18:13
  • @SEUNGMINSIM See theorem 1 in [Mező/Baricz 2017]. – IV_ Oct 03 '23 at 18:26
  • Okay, thank you so much. I will add something. thank you to tell me – SEUNGMIN SIM Oct 03 '23 at 18:53
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    Would you mind to have a look at https://math.stackexchange.com/questions/4923090/xx-x-1x1/4923276#4923276 ? I have the feeling that we are close to Lambert. Thanks and cheers. – Claude Leibovici May 31 '24 at 10:14