Answer
$\displaystyle\frac12y^2 + e^y = \frac12 x^2 - e^{-x} + C$
Work Step by Step
Separate the variables:
\begin{align*}
\frac{dy}{dx} &= \frac{x-e^{-x}}{y+e^y}\\[0.3cm]
(y+e^y) \, dy &= (x-e^{-x}) \, dx
\end{align*}
Now integrate both sides:
\begin{align*}
\int(y+e^y) \, dy &= \int(x-e^{-x}) \, dx\\[0.3cm]
\frac12y^2 + e^y &= \frac12 x^2 - e^{-x} + C
\end{align*}
Note that there is no way to completely isolate $y$ on one side. So our answer must be an implicitly defined function.
