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