$$
\frac{dy}{dx} = \frac{x - e^{-x}}{y + e^y}
$$
Rewrite the equation:
$$
(e^{-x} - x) + (y + e^y)\frac{dy}{dx} = 0
$$
The equation is of the form
$$ M(x) +N(y)\frac{dy}{dx} = 0 $$
Hence, it is separable.
$$
\int (y + e^y) dy = \int (x - e^{-x}) dx
$$
$$
\frac{y^2}{2} + e^y = \frac{x^2}{2} + e^{-x} + c'
$$
Let $$c = 2c'$$
Solution:
$$
y^2 - x^2 + 2(e^y - e^{-x}) = c
$$
where $$y + e^y \neq 0$$
