Suppose a cylinder of radius $R$ and length $L$, with longitudinal polarization $\vec{P}=P(r)\hat{z}$, where $(r,\theta,z)$ are usual cylindrical coordinates.
I want to compute the electric field $\vec{E}$ on the cylinder axis.
My reasoning is this: volumetric bound charge is zero, surface bound charge is $P(r)$ on one cap, $-P(r)$ on the other, so basically we have the field due to two charged disks, end of story.
I hesitate because I suspect the displacement field $\vec{D}$ should be useful, but I don't know how to compute it.
How do we compute the displacement field in this case?