Question

solve the differential equation
(x2- y2)DX +2xydy = 0
​

Answer 1

Answer

Given, (x²+y²)dx−2xydy=0

⇒(x²+y²)dx = 2xydy

⇒ dy/dx = x²+y²/2xy.... (i)

Let y=vx

Thus,  dy/dx =v+x dv/dx

​Thus, v+x dv/dx = x²(vx)²/2x+(vx)

⇒v+x dv/dx = 1 + v²/ 2v

⇒x dv/dx =  1 + v²/ 2v − v

⇒x dv/dx =  1 + v²/2v − v

⇒x dv/dx = 1 − v²/2v  

⇒ dx/x = 2v/1 −v² x dv

⇒ dx/x − 2v/1 −v² dv = 0  ....(ii)

Integrating both sides, we have

logx+log(1−v² )=logC

⇒logx(1−v²)=logC

⇒x(1−v²)=C

⇒x(1− y²/x²  )=C

⇒x( x² − y²/ x² )=C

⇒x²−y²  =Cx