The solution of the equation $(y+x\sqrt{xy}(x+y))dx-(x-y\sqrt{xy}(x+y))dy=0$ is

$x^2+y^2=4tan^{-1}\sqrt{\frac{y}{x}}+c$

The given equation can be written as

$x\,dx+y\,dy+\frac{ydx-xdy}{\sqrt{xy}(x+y)}=0$

or $\frac{1}{2}d(x^2+y^2)=\frac{ydx-xdy}{x^2\sqrt{\frac{y}{x}}(1+\frac{y}{x})}=\frac{2}{(1+\frac{y}{x})}d(\sqrt{\frac{y}{x}})$

$⇒x^2+y^2=4tan^{-1}\sqrt{\frac{y}{x}}+c$

Hence (B) is the correct answer.