Solution of differential equation $x dy-y dx=0$ represents :

family of straight lines passing through origin

$x d y-y d x=0$

$\Rightarrow \frac{x d y-y d x}{y^2}=0$   (integrating both sides) 

we get $\int \frac{x d y-y d x}{y^2}=\int 0$    

as $d(\frac{x}{y})$

$\frac{x d y-y d x}{y^2}=0$

$\Rightarrow \int d(\frac{x}{y})=\int 0$

$\Rightarrow \frac{x}{y}=c$

or  $y=k x$

family of straight lines passing through origin