Answer to Question #184608 in Differential Equations for Selva

Question

(Px-y)(Py+x)=2P

Answer 1

(px-y)(x-py)=2p

(pxy-y²)(x-yp)=2yp

(py/x.x²-y²)(1-py/x=2py/x

py/x.x ²-y²=[2py/x]/1-py/x

y²=py/x.x²-[2py/x]/[1-py/x]

then x²=u,y²=v

du=2xdx,dv=2ydy

p=dy/dx=2x/2y.dv/du=u/v.p,p=dv/du

py/x=dv/du=P

substitute

v=Pu-2P/1-P

v=Pu+ f(9)

v=Pu- f(p)

the solution is given by,

v=Cu-2C/1-C,c is arbitrary constant.