Are there any closed form solutions to $$ x \exp(x) = a x + b $$ for real-valued $x$, $a$, and $b$ (we wish to solve for $x$, and $a$ and $b$ are simple constants)? We can assume that everything is positive.

I know that the solution to $$ x \exp(x) = b $$ can be expressed via the Lambert $W$ function, and $$ x \exp(x) = a x $$ is a simple log, but I can't get anything out of there.

  • $\begingroup$ The solution to the last equation is simple log in addition to $x=0$. Don't forget that. $\endgroup$ – Arthur Aug 10 '16 at 22:09
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    $\begingroup$ I suspect no closed form using $W$, but cannot prove. Here is very tangentially related. $\endgroup$ – 6005 Aug 11 '16 at 5:48
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    $\begingroup$ If $b\approx0$, or close enough to $0$ where $a$ "dominates" the value of the LHS, then the solution is almost $x\approx\ln(a)$. If $a\approx0$, then the same argument gives $x\approx W(b)$. Not to much you can do if neither are the case. $\endgroup$ – Simply Beautiful Art Aug 17 '16 at 22:25

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