Complex analysis and real part of an analytic function

[Pouyan89]

Show that xu_x + yu_y is the real part of an analytic function if u(x,y) is.
To which analytic function is the real part of u = Re (f(z))?


What I know about analytic functions is Cauchy-Riemann condition
(∂u/∂x) =(∂v/∂y) and (∂y/∂y)=-(∂v/∂x)
If I say that F(z) = U(x,y) + iV(x,y)
If U(x,y) = x (∂u/∂x) + y(∂u/∂y)

Then by Cauchy-Riemann

(∂U/∂x) =(∂V/∂y) and (∂U/∂y)=-(∂V/∂x)

And :

∂V/∂x = (-∂u/∂y) - x(∂²/∂x∂y) -y(∂²/∂y²)

∂V/∂y = y(∂u/∂x) +x(∂²u/∂x²)+y(∂²u/∂x∂y)

I don't know how I can integrate it to find V. But am I in right path ?! Or is my algorithm completely wrong and I should find another way to show that ?!