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Find the integral curve of the differential equation $$x(1-x\ln y)\frac{dy}{dx}+y=0$$ which passes through $(1,1/e)$.

It looks like some king of substitution. I did try some exponential substitution but they were useless. Some hints are appreciated. Thanks.

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  • $\begingroup$ What about $y=e^u,dy=e^udu$? This results in $x(1-xu)\frac{du}{dx}=-1$, which is not obviously separable, but simpler. $\endgroup$ – Kajelad Aug 2 '16 at 17:23
  • $\begingroup$ @Kajelad yeah, that is what I tried. It does simplify, but it still gives problems. $\endgroup$ – user167045 Aug 2 '16 at 17:23
  • $\begingroup$ @User1upon0 after that substitution, you can separate variables and it solves easily after that. $\endgroup$ – D. Dmitriy Aug 2 '16 at 17:25
  • $\begingroup$ @D.Dmitriy could you please post a solution? $\endgroup$ – user167045 Aug 2 '16 at 17:25
  • $\begingroup$ @User1upon0 sorry, I've made a mistake. it's a little bit more involved than that. $\endgroup$ – D. Dmitriy Aug 2 '16 at 17:34
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Perform a change of variables $$ t=\ln{y}$$ $$z=\frac{1}{x}$$ Then the equation becomes $$\frac{dz}{dt}=z-t$$ This is a linear equation, you will be able to solve it by finding the formula online.

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