e^x-1/x^2 = 0

**munkaka** · December 2nd 2011, 11:32 AM

hi, i have to solve this equation:
$e^x-\frac{1}{x^2}=0$
it put it in wolframalpha and it says that the solution is $2W(\frac{1}{2})$ , 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

**wnvl** · December 2nd 2011, 12:03 PM

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

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