Given a Laplace equation with boundary conditions:
$\frac{\partial^2 u(x, y)}{\partial x^2}+\frac{\partial^2 u(x, y)}{\partial y^2}= 0 $, $(0\leq x, 0\leq y\leq 1)$
$\lim_{x\rightarrow \infty} u(x, y ) = 0$
$\frac{\partial u(x, y)}{\partial y}|_{y=0} = 0$
$u(x, 1) = 0$
$\frac{\partial u(x, y)}{\partial x}|_{x=0} = 1 + \cos\pi y$
When I tried solving this by assuming $u(x,y)= X(x)Y(y)$, we get by condition $1$ $X(x)= ae^{-kx}$ for some $k>0, a\in \mathbb R$, then by condition $2, 3$, we get $Y(y)=b\cos (ky)$ for some $b\in \mathbb R$ and $k = \frac{\pi}{2}+n\pi$ where $n$ could be $0, 1, 2, 3...$
But when considering condition 4, write $u=\sum c_nu_n (x, y)$ where $c_n\in \mathbb R$, $u_n(x,y) = e^{-k_nx}\cos(k_ny)$ with $k_n = \frac{\pi}{2}+n\pi$, this doesn't seem matching $1+\cos \pi y$, then I'm stucked.
Help please!