Solution of (d y)/(d x)=e^y+x+e^y-x is :

$\frac{d y}{d x}=e^y . e^x+e^y . e^{-x}=e^y\left(e^x+e^{-x}\right)$

$\Rightarrow e^{-y} dy=\left(e^{x}+e^{-x}\right) dx \Rightarrow e^{-y}=e^{x}-e^{-x}+c$

Hence (4) is the correct answer.