Solution of the differential equation $\frac{d x}{x}+\frac{d y}{y}=0$, is

$\frac{1}{x}+\frac{1}{y}=C$ $\log x \log y=C$ $x y=C$ $x + y=C$

$x y=C$

We have, $\frac{1}{x} d x+\frac{1}{y} d y=0$ On integrating, we get $\log x+\log y=\log C \Rightarrow x y=C$