$u ( x, y) = 2x (1− y)$ for all real $x$ and $y$ I am stuck on the following problem :  

Let $u ( x, y) = 2x (1− y)$ for all real $x$ and $y$.
  Then a function v ( x, y), so that
  $f ( z) = u ( x, y) + iv (x, y)$ is analytic, is   
(a) $x^2-(y-1)^2$  
(b) $x^2+(y-1)^2$  
(c) $(x-1)^2-y^2$  
(d) $(x-1)^2+y^2$   

My Attempt: For analytic function $f(z)$, it must satisfy the Cauchy-Riemann equation.
$u_x=v_y \implies v_y=2(1-y) \implies v=2y-y^2 +c$. Now,I am stuck. Can someone help?

Hint: you have a slight mistake in your reasoning. The partial with respect to $v_{y} = 2(1-y) \implies v = 2y - y^{2} + h(x)$... using this, which of the answers makes sense?
