I'm sure this is a very basic differential equation but I cant figure out how to separate it to begin with.
$\displaystyle \frac{dy}{dx} = e^{x + y} + e^x\sin{x}$
You're welcome. Incidentally, I don't think you'll be able to find a closed-form solution of this DE. That is, you'll still have integrals in your final solution, because there are no known antiderivatives for the integrals you get. That's ok. You still get an explicit solution (all the x's on one side), which is nice.
Oh sorry I mistyped the problem in the OP, it's actually:
$\displaystyle \frac{dy}{dx} = e^{x + y} + e^y\sin{x}$
from which I get...
$\displaystyle \int \frac{dy}{e^{2y}} = \int e^x + sinx dx$
$\displaystyle \frac{e^{-2y}}{2} = e^x - cosx$
and finally...
$\displaystyle 2y = \ln{\cos{x}} - x$
Is that correct?