Show hat if $v(x,y)$ is harmonic conjugate of $u(x,y)$ in a domain $D$. Then every harmonic conjugate of $u(x,y)$ must be of form $v(x,y)+a$ where a is real constant
My idea:
since $v$ is harmonic conjugate of $u$ then it satisfies C-R equation
then $u_x=v_y,v_x=-u_y$ so result comes from this equation is i am right?
Yes, you are right. It follows from your observation that if you have two harmonic conjugates of the same function, then they have the same partial derivatives with respect to both variables. Therefore, their difference is constant (and real, of course, since we are dealing with real functions here).