Let $u: \mathbb R^2 \to \mathbb R$ be a function that satisfies
Laplace equation $u_{xx}(x,y) + u_{yy}(x,y) = 0$ for $x \in [0,\pi]$ and $y \in [0,1]$
$u_{x}(0,y)=0, y \in [0,1]$
$u_x(\pi,y)=0, y \in [0,1]$
$u(x,0)=a^2 \cos(x), x \in [0,\pi]$, for some $a \ne 0$ (I think $a$ is a real constant)
$u(x,1)=a^2 \cos^2(x), x \in [0,\pi]$
Question 1: If there exist $F, G:\mathbb R \to \mathbb R$ such that $u(x,y)=F(x)G(y)$ for all $x \in [0,\pi]$ and $y \in [0,1]$, then what are possible $u$'s?
I don't think there are any because $u(x,0)=a^2 \cos(x) = F(x)G(0)$ and $u(x,1)=a^2 \cos^2(x) = F(x)G(1)$ give us $a^2 \cos^2(x) = F(x)G(0) \cos(x) = F(x)G(1)$. It seems that $F(x)G(0) \cos(x) = F(x)G(1)$ makes sense if and only if $G(0)=G(1)=0$ but then with $a^2 \cos^2(x) = F(x)G(0) \cos(x)$, we would get $a=0$.
Also I've solved $\frac{F''(x)}{F(x)} = \frac{-G''(y)}{G(y)} = -k$ for the cases of $k < 0$, $k = 0$ and $k > 0$, and I think $k<0$ and $k=0$ end up contradicting conditions, so we must have $k>0$. However, I think $k>0$ might have some contradictions too. I don't want to type up what I got for $F$ or $G$ without first resolving the issue in the previous paragraph.
Question 2: If there exist $F, G:\mathbb R \to \mathbb R$ such that $u(x,y)=F(x)G(y)$ for all $x \in [0,\pi]$ and $y \in [0,1]$, then in solving for the possible $u$'s if any, are complex variables needed? I know that for analytic/holomorphic functions $f: \mathbb C \to \mathbb C$, $f(z)=v(x,y)+iw(x,y), z=x+iy$, we have $v$ and $w$ harmonic and thus solutions of Laplace equation, but I don't quite see the relevance here.
Question 3: Can someone please provide a reference on obtaining general solutions to this kind of problem with these kinds of conditions?
I believe it begins with considering $u$ as separable into $u=FG$ and then solving for $u$ in general just like with the heat and wave equation and then you get to Fourier series, but I was not able to find anything explicit in Pinchover and Rubinstein (It's been several years since I've read this book. I mostly read heat and wave. I'm learning Laplace for the first time now.) or this link that directly and completely helps. In particular, I don't think either the book or the link contain examples for the kinds of conditions above.
This is the closest kind of problem I've seen on stackexchange. It has $u(x,0)=0$ though. This leads me to suspect such problems might be a little advanced.
Answer: This question appears to have been answered with the comments below. Please now refer to my new question: 2D Laplace equation with two initial conditions