Getting started on first order differential equation
Solution 1:
$\dfrac{dy}{dx}=\dfrac{x-e^y}{y+e^y}$
$(x-e^y)\dfrac{dx}{dy}=y+e^y$
The substitution $u=x-e^y$ brings the above ODE to the Abel equation of the second kind of the form $u\dfrac{du}{dy}=y+e^y-e^yu$
The substitution $u=\dfrac{1}{v}$ brings the above ODE to the Abel equation of the first kind of the form $\dfrac{dv}{dy}=e^yv^2-(y+e^y)v^3$
http://www.hindawi.com/journals/ijmms/2011/387429/#sec2 claims that it has analytical method to solve this Abel equation of the first kind.