If `(x+i y)^3=u+i v ,`then show that `u/x+v/y=4(x^2-y^2)`.

Explanation

`(x+iy)^(3)= u + iv`
`or x^(3) + (iy)^(3) + 3.x.iy (x+iy) = u + iv`
`or x^(3)+ i^(3)y^(3) + 3x^(2) yi + 3xy^(2)i^(2) = u + iv`
`or x^(3) - iy^(3) + 3x^(2)yi - 3xy^(2)= u + iv`
` or (x^(3)- 3xy^(2)) + i(3x^(2)y - y^(3)) = u + iv`
On equating real and imaginary parts, we get
`u = x^(3) - 3xy^(2), v = 3x^(2)y-y^(3)`
`therefore (u)/(x) + (v)/(y) = (x^(3)-3xy^(2))/(x)+ (3x^(2)y-y^(3))/(y)`
`= (x(x^(2)-3y^(2)))/(x)+(y(3x^(2) - y^(2)))/(y)`
`= x^(2) - 3y^(2) + 3x^(2) -y^(2)`
`= 4x^(2)-4y^(2) =4(x^(2)-y^(2))`