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 \varphi² /dx² +Acos(x)cos(y) d \varphi² / 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?