# Getting started on first order differential equation

I'm trying to solve this differential equation:

$$\frac{dy}{dx} = \frac{x-\exp(y)}{y+\exp(y)}$$

I thought I could use separation of variable, but I'm unable to isolate $x$.

Could you please help me get started on this differential equation?

Thank you

• I don't think this differential equation has a closed form solution... – Zen Jun 9 '13 at 5:44
• $x=-y$ appears to be a solution ... – Mark Bennet Jun 9 '13 at 5:49
• My bad - I should have said a closed form general solution. It definitely is not separable. – Zen Jun 9 '13 at 6:00

## 1 Answer

$\dfrac{dy}{dx}=\dfrac{x-e^y}{y+e^y}$

$(x-e^y)\dfrac{dx}{dy}=y+e^y$

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.