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Solve x^3*e^(-a/x)=b

  1. Nov 1, 2006 #1
    How would you solve for x algebraically?

    [tex]x^3 e^{\frac{-a}{x}} = b [/tex]

    where a and b are some constants.
  2. jcsd
  3. Nov 1, 2006 #2


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    Staff: Mentor

    Is this homework? If so, I can move it.

    Start by isolating the logarithmic terms and the non-log terms. What can you do to get the x^3 away from the e^ term? Once you do that, what can you do to both sides of the equation to get rid of the e^?
  4. Nov 1, 2006 #3


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    Homework Helper

    The Lambert W function is the first thing to try when you have something that looks like that. It has the property that W(x e^x) = x. So try to rearrange that into the form f(x) e^f(x) = C for some constant C, and then apply W to both sides to get f(x)=W(C).
  5. Nov 1, 2006 #4
    I don't think that's gonna help. If you do that, then you are just trapping x inside the natural log function instead of the exponential function.
  6. Nov 1, 2006 #5
    Wow! That's news to me. I searched the lambert W function on Wikipedia and I have to say it is pretty interesting. Let me see what I can do...
  7. Nov 1, 2006 #6
    Got it!. I am 99% sure that the answer is:
    [tex] x = \frac{a}{3W(\frac{ab^{-1/3}}{3})} [/tex].
    Last edited: Nov 2, 2006
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