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Can anyone please help me with a close-form explicit solution to $y\times\ln(y)=x\times\exp(x)$ ? Please note that $y=\exp(x)$ is not the answer I am looking for. The equation actually looks like this $y=\exp(x/y*\exp(x))$

hyper-neutrino
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Song Li
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    Why not $y = e^{x}$?. If you take $y = e^{x}$, then

    $yln(y) = e^{x}ln(e^{x}) = e^{x}x$

     – ZAF Nov 24 '21 at 19:10
  • This is the spiral equation created by picking those points on the spiral y=exp(x) with equal interval, say, one point per period. The physics is R=R0exp(ktheta) where theta =omega*time. To simply, I let R0=1, k=1. – Song Li Nov 24 '21 at 19:20
  • "$y=\exp(x)$ is not the answer I am looking for": why do you reject a correct answer ??? –  Nov 24 '21 at 19:27
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    I think it's perfectly reasonable to ask for solutions which are not obvious or not already known. In math we ask for nontrivial solutions all the time so I'm confused by the comments to this question. – Chris Brooks Nov 24 '21 at 19:30
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    You would get better feedback by explaining this first. We can't read your mind. –  Nov 24 '21 at 19:41

1 Answers1

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With $y=e^t$, the equation reads

$$te^t=xe^x.$$

An obvious solution is $t=x$. Anyway, for $t<0$, the LHS function is not invertible, and you need the two branches of Lambert's function.

$$t_0=W(xe^x)=x,\\t_1=W_{-1}(xe^x).$$