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Thread: e^x-1/x^2 = 0

  1. #1
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    Dec 2011
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    e^x-1/x^2 = 0

    hi, i have to solve this equation:
    $\displaystyle e^x-\frac{1}{x^2}=0$
    it put it in wolframalpha and it says that the solution is $\displaystyle 2W(\frac{1}{2})$, where W is the Lambert W-function, but i dont know how it find this. Can someone help me to show that the solution is really that? thanks
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  2. #2
    Junior Member
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    Oct 2011
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    Re: e^x-1/x^2 = 0

    $\displaystyle e^x-\frac{1}{x^2} = 0$
    $\displaystyle e^x=\frac{1}{x^2}$
    $\displaystyle x=-2ln|x|$
    $\displaystyle \frac{x}{2}=-ln|x|$
    $\displaystyle e^{x/2}=\pm \frac{1}{x}$
    $\displaystyle \frac{x}{2}.e^{\frac{x}{2}}=\pm \frac{1}{2}$
    $\displaystyle \frac{x}{2}.e^{\frac{x}{2}}=\pm \frac{1}{2}$
    $\displaystyle \frac{x}{2}=W(\pm \frac{1}{2})$
    $\displaystyle x=2W(\pm \frac{1}{2})$
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