Question Number 6177 by FilupSmith last updated on 17/Jun/16
$${f}\:'\left({x}\right)={xf}\left({x}\right) \\ $$$${f}\left({x}\right)=?? \\ $$
Answered by Yozzii last updated on 17/Jun/16
$$\frac{{df}}{{dx}}={xf} \\ $$$$\Rightarrow\int\frac{{df}}{{f}}=\int{xdx} \\ $$$${ln}\mid{f}\left({x}\right)\mid=\frac{{x}^{\mathrm{2}} }{\mathrm{2}}+{C} \\ $$$${f}\left({x}\right)={Ae}^{{x}^{\mathrm{2}} /\mathrm{2}} \:\:\left({A}={arbitrary}\:{constant}\right) \\ $$$${f}'\left({x}\right)={xAe}^{{x}^{\mathrm{2}} /\mathrm{2}} ={xf}\left({x}\right) \\ $$