Differential Equations and Linear Algebra (4th Edition)

Using exponent rules, $e^{x+y}$ can be changed to $e^xe^y$. This changes the equation to $$\frac{dy}{dx}=e^xe^y$$ The variables can be separated by multiplying the entire equation by $\frac{dx}{e^y}$. $$\frac{dy}{e^y}=e^xdx$$ Therefore, the equation is separable.