TITLE: Factoring $y$ out of $y^2 + e^y$ possible?
QUESTION [5 upvotes]: I'm working on solving a separable first order differential equation where I get down to:
$$\frac12 y^2 + e^y = \frac12x^2 + e^{-x} + c$$
Is it possible to solve for $y$ here?  I can't think of a way to factor the left hand side.
Here's the original differential equation:
$$\frac{dy}{dx} = \frac{x-e^{-x}}{y + e^y}$$
Thanks!

REPLY [2 votes]: Apparently answered in comments:

Nope, not really. It's clear $y=−x$ is a solution for $C=0$. Perhaps you'd
  get more useful responses if you posted the original problem you're
  working on.. – anon Apr 22 at 23:31
No, you cannot factor $y$ from the left hand side. What is the
  differential equation you were trying to solve? – Arturo Magidin Apr
  22 at 23:31
Ok I posted the original problem. Don't think I made a mistake
  elsewhere but possible. – dacc Apr 22 at 23:34
Are there no initial conditions? Otherwise it looks like you'll have
  to be content with this implicit solution. – anon Apr 22 at 23:39
ok thanks, guess that's it. @sdcvvc ah thanks for catching that sign –
  dacc Apr 22 at 23:41
Hm, I don't think what I said was relevant. It seems anon is right and
  either you have a condition giving $C=0$, or the implicit solution is
  all what you can get... – sdcvvc Apr 22 at 23:45