The solution of the differential equation xdy−ydx=x2+y2dx is

dydx=y+x2+y2x=yx+x2+y2xdydx=yx+x1+yx2xdydx=yx+1+yx2put y=vxdydx=v+xdvdxv+xdvdx=v+1+v2xdvdx=1+v2⇒∫dv1+v2=∫dxx⇒lnv+v2+1=log⁡x+log⁡c⇒ln⁡yx+yx2+1=ln⁡|cx|⇒y+x2+y2=cx2