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$\frac{d}{dt}\left(\frac{x'(t)}{x(t)}\right)=x(t)-x^2(t)$ where $x'(t)=\frac{d}{dt}x(t)$ What reasoning (if it exists) I can apply to solve this differential equation? Thanks.

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    A good motivation for Julián's method is to notice that the $\frac{x'}{x}$ term is just $(\ln x)'$, so the desired function is the exponential of another.2012-07-04

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Let $x=e^u$; then $u$ satisfies de differential equation $ u''=e^u-e^{2u}. $ Multiply by $u´$ and integrate once to get $ \frac12(u')^2=e^u-\frac12\,e^{2u}+C_1. $ This is a differential equation in separeted variables, whose solution is $ \int\frac{du}{\sqrt{2\,e^u-e^{2u}+2\,C_1}}=\pm t+C_2. $

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    Oh, I see. I was speculating that, but I wasn't sure. Thank you Julian!2012-07-06