Let y =vx
=>dy/dx=v+xdv/dx
Now given equation is
x^2dy+(xy+y^2)dx=0
=> dy/dx=-y/x -(y/x)^2
=>v+xdv/dx=-v-v^2
=>xdv/dx=-v^2-2v
=>dx/x=dv/(-v^2-2v)
=>dx/x=-dv/(v(v+2))
=>dx/x=1/2[dv/(v+2)-dv/v]
Integrating we get
=>2logx=log(v+2)-logv +logc
=>log[(vx^2)/(v+2)]=logc
=>xy/((y/x+2)=c
=>x^2y/(2x+y)=c