Question

(x²-2xy-y²)dx-(x+y)²dy=0​

Answer 1

Answer:

(x2+y2)dx+2xydy=0⇒dydx=−x2+y22xy

This is homogeneous equation

Let,y=vx⇒dydx=v+xdvdx⇒v+xdvdx=−x2+v22x.vx⇒xdvdx=−v−1+v22v⇒xdvdx=−1+3v22v⇒dxx=−2v1+3v2⇒∫dxx=−∫2v1+3v2⇒logx=−13log(1+3v2)+logc⇒logx+13log(1+3v2)=logc⇒logx(1+3v2)13=logc⇒x(1+3v2)13=c⇒x(1+3y2x2)13=c⇒x(x2+3y2x2)13=c