The solution of the differential equation $(x+y)(d x-d y)=d x+d y$, is

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

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

We have, $(x+y)(d x-d y)=d x+d y$ $\Rightarrow d x-d y=\frac{d x+d y}{x+y}$ $\Rightarrow d(x-y)=\frac{d(x+y)}{x+y}$ $\Rightarrow x-y=\log (x+y)+\log C$                 [On integrating] $\Rightarrow c(x+y)=e^{x-y}$ $\Rightarrow x+y=k e^{x-y}$, where $k=\frac{1}{C}$