If (x + iy)3 = u+ iv, then show that u/x + v/y = 4(x2 - y2)
Solution:
Given,
(x+iy)3= u + iv
or, x3 + 3x2yi + 3xi2y2 + i3y3= u + iv
or, x3 – 3xy2 + 3x2iy – iy3= u + iv
or, x(x2 – 3y2) + y(3x2 – y2)i= u+iv
Comparing Corresponding Terms,
u = x(x2 – 3y2) and v = y(3x2 – y2)
Now,
LHS = u/x + v/y
= (x(x2 – 3y2))/x + (y(3x2 – y2))/y
= (x2 – 3y2) + (3x2 – y2)
= 4x2 – 4y2
= 4(x2 – y2)
= RHS
Proved.......