Electrodynamics: Electrostatic field potencial in Cartesian coordinates

[C12H17]

Homework Statement

It's given that absolute permitivity is a coordinate function: ε (x, y, z) = Asin(x)cos(y), where A=const

Homework Equations

We need to find an electrostatic field potential function $\varphi$ in Cartesian coordinate system.

The Attempt at a Solution

I tired to solve, but I don't know if it's ok. Check, please?

$\vec{D}$=ε$\vec{E}$ and $\vec{E}$= - grad$\varphi$
div$\vec{D}$=ρ
ρ=$\frac{dDx}{dx}$+$\frac{dDy}{dy}$+$\frac{dDz}{dz}$
$\vec{E}$=$\vec{x}$₀εE_x+$\vec{y}$₀εE_y+$\vec{z}$₀εE_z
D_x=εE_x
D_y=εE_y
D_z=εE_z
then
ρ=d Asin(x)cos(y)E _x / dx + dAsin(x)cos(y)E _y /dy + d Asin(x)cos(y)E_z / dz=Asin(x)sin(y)$\frac{dE}{dx}$+Acos(x)cos(y)$\frac{dE}{dy}$+Asin(x)cos(y)$\frac{dE}{dz}$=Asin(x)sin(y) d $\varphi$² /dx² +Acos(x)cos(y) d $\varphi$² / dy² + Asin(x)cos(y) d $\varphi$ ²/ dz²

and now i don't know. ;D

 

[rude man]

You're not given any charge distribution, any boundary conditions?