# Solve the differential equation using substitution

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.

• 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. – Kajelad Aug 2 '16 at 17:23
• @Kajelad yeah, that is what I tried. It does simplify, but it still gives problems. – user167045 Aug 2 '16 at 17:23
• @User1upon0 after that substitution, you can separate variables and it solves easily after that. – D. Dmitriy Aug 2 '16 at 17:25
• @D.Dmitriy could you please post a solution? – user167045 Aug 2 '16 at 17:25
• @User1upon0 sorry, I've made a mistake. it's a little bit more involved than that. – D. Dmitriy Aug 2 '16 at 17:34

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.