The problem: Find the harmonic conjugate of $G(x,y)= 2v^2(x,y)-2u^2(x,y)$
My attempt to solving it
I know that
"If two given functions u and v are harmonic in a domain D and their first-order partial derivatives satisfy the Cauchy–Riemann equations throughout D, then v is said to be a harmonic conjugate of u."
So having $G(x,y)$ I'm searching for a function let's call it $H(x,y)$ that is the harmonic conjugate of $G(x,y)$. Applying the Cauchy-Rieman equations I'd need to fulfill these conditions:
$G_x=H_y$ and $G_y=-H_x$
I'm stuck at this point. I know how to do this if I am given a defined function, but not something as $v(x,y)$ or $u(x,y)$. If I continue with the "normal" method of solving these problems (when functions are defined) I'd derivate G in terms of either x and y and then integrate in terms of the other variable and so on. But for example: If I derivate G in terms of x I'd get:
My idea would be to integrate $G_x$ in terms of x... But that's where I get stuck.
I don't know if I'm missing some property that could help me in this case.
Any help will be much appreciated