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Algebra: Is this possible to solve?

  1. Aug 12, 2010 #1
    I came across this type of equation, and somehow I could not figure out how to solve it analytically. I ended up solving it numerically, but now I'm bothered and I want to know if this is possible.

    a = (1-exp(b/x) / (1-exp(c/x))

    a,b,c, are constants, x is the unknown
     
  2. jcsd
  3. Aug 12, 2010 #2
    It cannot be solved analytically for general numbers b and c.

    EDIT:

    suppose you introduce a new variable:

    [tex]
    y \equiv \exp\left(\frac{b}{x}\right)
    [/tex]

    Then, for the other exponential, you would have:
    [tex]
    \exp\left(\frac{c}{x}\right) = \exp\left(\frac{c}{b} \, \frac{b}{x}\right) = \left[\exp\left(\frac{b}{x}\right)\right]^{\frac{c}{b}} = y^{c/b}
    [/tex]

    and the equation becomes:

    [tex]
    a = \frac{1 - y}{1 - y^{c/b}}
    [/tex]

    [tex]
    y = 1 - a ( 1 - y^{c/b})
    [/tex]
     
    Last edited: Aug 12, 2010
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