अवकल समीकरण `(x^2 - y^2) dx + 2xy dy = 0` का हल है

Explanation

`2xy dy = (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)`
माना `y = vx implies (dy)/(dx)=v+x(dv)/(dx)`
`implies x(dv)/(dx)=(v^(2)-1)/(2v)-v=(-(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) implies x^(2)+v^(2)x^(2)=cx`
`implies x^(2)+y^(2)=cx`