I need to solve $\nabla^2 V(x,y)$ for $$\begin{cases} \text{$V \to 0$ when $x \to \infty$} \\\text{$V = 0$ when $y = 0$}\\\text{$V = 0$ when $y = a$}\\\text{$V = V_0$ when $0 \lt y \lt \dfrac a2, x = 0 $}\\\text{$V = -V_0$ when $\dfrac a 2 \lt y \lt a, x = 0 $}\end{cases}$$
$$\dfrac{\partial^2 V}{\partial x^2} + \dfrac{\partial^2 V}{\partial y^2} = 0$$
Using separation of variables, $V = X(x)Y(y)$
I got,
$$X^{\prime\prime}_x = k^2 X(x), Y^{\prime\prime}_y = -k^2 Y(x)$$
I solved these, $$V(x,y) = (Ae^{kx} + B e^{kx} )(C\sin ky + D \cos ky)$$
Using first three constraints, I got
$$V(x,y) = \sum^{\infty}_{n = 1}C_ne^{-n\pi x/a}\sin \left(\frac{n\pi y}{a}\right) \text{ for } n \in \Bbb Z_+$$
$$V(0, y) = \sum^{\infty}_{n = 1}C_n\sin \left(\frac{n\pi y}{a}\right) = \begin{cases}-V_0 \qquad \dfrac a 2 \lt y \lt a \\ V_0 \qquad 0 \lt y \lt \dfrac a 2\end{cases}$$
At this point I am confused as how to find $C_n$. I thought I will multiply by $\sin (\frac{m\pi y}{a})$ and integrate from $0$ to $\frac{a}{2}$ for bottom cases and likewise $\frac{a}{2}$ to $a$ for top case but I think that will give me two values for same $C_m$, which can't true. What should I do ?
Edit : $$ \sum^{\infty}_{n = 1}C_n \int_0^{a} \sin\left(\dfrac{m\pi y}{a}\right) \sin\left(\frac{n\pi y}{a}\right)dy = \int_0^{a} V(y) \sin\left(\dfrac{m\pi y}{a}\right)dy = V_0\left(\int_0^{a/2}\sin\left(\dfrac{m\pi y}{a}\right)dy - \int_{a/2}^{a}\sin\left(\dfrac{m\pi y}{a}\right)dy\right) = \dfrac{V_0a}{\pi m}\left(-\cos(\pi m /2) + 1 +cos (m\pi) -\cos\left( \dfrac{m\pi}{2}\right)\right) = \dfrac{V_0 a}{\pi m}\left(1-2\cos\left(\dfrac{m\pi}{2}\right)+\cos(m\pi)\right)$$
$$\therefore C_m = \dfrac{8V_0}{\pi m}$$ for odd $n$ where $m = 2n$.
Is this correct ?
You can find $C_n$ multiplying by $\sin(\frac{m\pi y}{a})$ and integrating from $0$ to $a$: define $$ V_0(y)= \begin{cases} V_0, & 0<y<\frac{a}{2} \\ -V_0, & \frac{a}{2}<y<a \end{cases} $$
knowing that $$ \int_0^a \sin(\frac{n\pi y}{a}) \sin(\frac{m\pi y}{a}) dy= \begin{cases} 0, & \text{if $n\neq m$} \\ \frac{a}{2}, & \text{if $n= m$} \end{cases} $$ you arrive at $$ C_n=\frac{2}{a}\int_0^a V_0(y) \sin(\frac{n\pi y}{a}) dy $$ now you just split the integral from $0$ to $\frac{a}{2}$ and from $\frac{a}{2}$ to $a$ and integrate; $C_n$ will be zero for some $n$ and non-zero for others (e.g. $C_2$).