The general solution of the differential equation

$\frac{d y}{d x}=e^{x+y}=e^{x} \cdot e^{y}$

$\Rightarrow \frac{d y}{e^{y}}=e^{x} d x$

$\Rightarrow e^{-y} d y=e^{x} d x$

Integrating both sides, we get:

$\int e^{-y} d y=\int e^{x} d x$

$\Rightarrow-e^{-y}=e^{x}+k$

$\Rightarrow e^{x}+e^{-y}=-k$

$\Rightarrow e^{x}+e^{-y}=c$                  $(c=-k)$

Hence, the correct answer is A.