Solve $$(1 - 2e^{x/y})dx -2e^{x/y}(1 - \frac{x}{y})dy = 0$$
Even though this is not a homogeneous expression I tried using the substitution $x =ty$ which led me nowhere conclusive. Next I tried to use the derivative of $2e^{x/y}$ that simplified a couple of terms but the remaining bit was still complex.
Since $dx=tdy+ydt$, $$0=(1-2e^t)(tdy+ydt)-2e^t(1-t)dy=(t-2e^t)dy+y(1-2e^t)dt.$$Define $f=t-2e^t$ so $0=fdy+ydf=d(fy)$, i.e. $fy$ is constant. Thus$$y=\frac{-2y_0}{t-2e^t}.$$