Electric potential plane with periodic surface charge density

let a plane xOy with a surface charge density $σ(x)= σ_0 + σ_1cos(frac{2πx}{a}):::: σ_1,σ_0$ and a being positive constants

the electric potential has the form $V(x,y) = V_0(z)+V_1(z)cos(frac{2πx}{a})$

using Laplace’s equation find $V_0(z)$ and $V_1(z)$

we have $Delta V = frac{partial^2 V}{partial x^2}+frac{partial^2 V}{partial y^2}+frac{partial^2 V}{partial z^2}=-V_1(z)(frac{2π}{a})^2cos(frac{2πx}{a})+frac{d²V_0(z)}{mathrm{dz²}}+frac{d²V_1(z)}{mathrm{dz²}}cos(frac{2πx}{a})=0$ and I can’t go any further.