Are there any closed form solutions to
$$(ax+b)\exp(x)= cx+d$$
for real-valued $a,b,c$ and $d$?
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The solution is by the function you mentioned: Lambert $W,.$ – Kurt G. Sep 21 '23 at 10:02
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2I do not see a solution of this with Lambert W. However, you can solve $(ax+b)\exp(x) = d$ with Lambert W. You can also solve $b\exp(x) = cx+d$. – GEdgar Sep 21 '23 at 10:05
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@GEdgar. It is the generalized Lambert function – Claude Leibovici Sep 21 '23 at 10:08
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@njuffa. Done. Nice but not very practical (from my point of view). Cheers – Claude Leibovici Sep 21 '23 at 10:25
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Aside from the series representation in the Arxiv paper, there is a way to get a Mellin Barnes type solution and a real integral representation like here – Тyma Gaidash Sep 21 '23 at 12:01
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Rewrite the equation as $$e^{-x}=\frac{a x+b}{c x+d}=\frac a c\,\frac{x+\frac b a}{x+\frac d c}$$
The solution is given in terms of the generalized Lambert function (have a look at equation $(4)$ in the paper).
Claude Leibovici
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István Mező and Árpád Baricz, "On the generalization of the Lambert $W$ function." Transactions of the American Mathematical Society, Vol. 369, No. 11, Nov. 2017, pp. 7917-7934 (online) – njuffa Sep 21 '23 at 10:27
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