Find all harmonic functions $u(x,y)$ in the unit disk $x^2+y^2<1$ such that $u_y=0$ for $x^2+y^2<1$. What can be said about $u(x,y)$?
I'm not sure how to solve this problem. I think we can set $f(z)=u+iv$, an analytic function then by Cauchy Riemann, we get $v_x=0$, so $v=\phi(y)+c$. From which we get $v_y=\phi'(y)=u_x$ so $u=x\phi'(y)+c$. But $u_y=x\phi''(y)=0$ so $\phi''=0$ and $\phi'=k$ a constant. This gives $u=xk+c$, where $k$ and $c$ are constants.
Is this the right way to solve this problem? I would greatly appreciate any help.
You're definitely on the right track, but no need for Cauchy-Riemann.
Since $u_y=0$, we have $u(x,y)$ is only a function of $x$, say $u(x,y) =f(x)$. Next,
$$u_y= 0\Rightarrow u_{yy}=0\Rightarrow u_{xx}=0.$$
(last implication because $u$ is harmonic).
Thus $u_x$ is only a function of $y$, say, $u_x = g(y)$. Therefore
$f(x) = u(x,y) = x g(y) + C$.
Since LHS is independent of $y$, it follows that $g$ is a constant function. Hence $f(x) = C'x+C$, where $C'$ and $C$ are (real) constants.