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Eh? How do I solve xe^(a/x) = b?

  1. Oct 2, 2008 #1
    Eh??? How do I solve xe^(a/x) = b?

    1. The problem statement, all variables and given/known data

    xe^(-a/x) = b, where a and b are numbers that are given and I'm trying to solve for x.

    3. The attempt at a solution

    All I can think of are Taylor series, which won't work in this case because a is ~4000 and b = ~0.1, so I'd need to expand to a lot of terms.

    I could try to do it numerically, but would rather not... I know the answer ends up being ~520.
     
  2. jcsd
  3. Oct 3, 2008 #2

    gabbagabbahey

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    Re: Eh??? How do I solve xe^(a/x) = b?

    You have a transcendental equation here, with no algebraic solution. Are you sure this is the correct equation?
     
  4. Oct 3, 2008 #3
    Re: Eh??? How do I solve xe^(a/x) = b?

    Yes. I'm trying to solve the equation that's in the book.

    It's for charge carrier density in a semi-conductor

    n^2 = B*T^3*exp(-E/kT)

    Where n, B, E, and k are constants that are given. So I divide by B, then cube root it to obtain T*exp(-a/T) = b

    A similar problem is later on in the homework set, where the professor says to calculate it numerically. I mean, the problem is identical except you first have to find n with another equation, then you're back to that. But he makes no mention of calculating it numerically here.

    I think, though, that he still wants us to, because like you said, no easy solution. It would take a team of Russian mathematicians in one of Russia's finest Gulag's at least a year to find the solution*. I'll just do this numerically also. In the least I needed confirmation that it's not a trivial answer. Thanks.

    *I'm taking a shot in the dark here.
     
  5. Oct 3, 2008 #4

    Gib Z

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    Re: Eh??? How do I solve xe^(a/x) = b?

    There is actually no possible way of expressing that number in terms of commonly known constants. It would only take, however, someone with a big ego typing onto Physicsforums to define, eg, The GibZ Constant, whose definition is the unique solution to the above equation, numerically approximately *blah blah*. Those are your options, but in the end, for a "useful" answer you'll need a numerical approximation, no way around it.
     
  6. Oct 3, 2008 #5
    Re: Eh??? How do I solve xe^(a/x) = b?

    Found an answer:

    http://en.wikipedia.org/wiki/Lambert_W_function

    Too bad it's useless... and still amounts to a numerical computation. Oh well. Thanks guys.
     
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