# Non-linear second-order ODE, $y''+y^{-1}(y')^2=y^{-1}e^{-y}y'$

I am not sure what method to use for the following ODE,

$$y''+y^{-1}(y')^2=y^{-1}e^{-y}y'$$

With $$v=y'$$, I got $$v'+y^{-1}v^2=y^{-1}e^{-y}v$$ and with $$w=v^{-1}$$,

$$w'+y^{-1}e^{-y}w=y^{-1}$$

$$y''+y^{-1}(y')^2=y^{-1}e^{-y}y'$$ $$yy''+(y')^2=e^{-y}y'$$ You have derivatives on both sides: $$(y'y)'=-(e^{-y})'$$ Integrate both sides.