Show that the given differential equation is homogeneous and solve each of them.`(x^2-y^2)dx+2x ydy=0`

Explanation

`2xydy=(y^(2)-x^(2))dx`
`implies (dy)/(dx)=(y^(2)-x^(2))/(2xy)`
`implies v+x(dv)/(dx)=(v^(2)x^(2)-x^(2))/(2x^(2)v)`
Let `y=vx`
`implies (dy)/(dx)=u+(dv)/(dx)`
`implies x(dv)/(dx)=(v^(2)-1)/(2v)-v=(-1(1+v^(2)))/(2v)`
`implies(2v)/(1+v^(2))dv=-(dx)/(x)`
`implies int(2v)/(1+v^(2))dv=-int(dx)/(x)+logc`
`implies log(1+v(2))=-logx+logc`
`implies 1+v^(2)=(c )/(x)`
`impliesx^(2)+v^(2)x^(2)=cx`
`implies x^(2)+y^(2)=cx`