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Given that $f:\mathrm{\Re }\to \mathrm{\Re }$ is invertible, differentiable and monotonic increasing.
Need to find a general solution for the equation:
${f}^{\prime }\left(y\right){y}^{\prime }=xf\left(y\right)$
I'm working to either separate variables or to put the equation in linear form of $y$ but no success so far.
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Hint:
${f}^{\prime }\left(y\right){y}^{\prime }=xf\left(y\right)$
$\frac{{f}^{\prime }\left(y\right)}{f\left(y\right)}dy=x\phantom{\rule{mediummathspace}{0ex}}dx$
$\mathrm{ln}|f\left(y\right)|=\frac{{x}^{2}}{2}+\text{constant}$
$f\left(y\right)=C\phantom{\rule{mediummathspace}{0ex}}{e}^{{x}^{2}/2}$
$y={f}^{-1}\left(C\phantom{\rule{mediummathspace}{0ex}}{e}^{{x}^{2}/2}\right)$
You have to justify the validity of the main steps.