Assumptions:

Cylindrical symmetry is valid because both cylinders are infinitely long.
The conducting shell is initially electrically neutral.
The cylinders are treated like long line of charges.

Step 1: Define the coordinate system.

A cylindrical coordinate system shall be used. The origin shall be from the common center of the two concentric cylinders.

To illustrate the regions of concern, refer to the image below.

Step 2: Identify the electric fields and the distribution of charges.

Looking at the layers, the electric fields would be like these

The nonconducting cylindrical shell with charge density

polarizes the conductor. The outer surface of the conductor becomes positively charged while the inner surface becomes negatively charged. Thus, we have three charged layers. These are P, P', and P'' with charge densities respectively.

Also,

Since we treat the long cylinders to be line of charges, we could define the charge density as

And, using Gauss's law, the electric field at distance r from the infinitely long line of charges is

Step 3: Solve for the electric fields.

At region I,

At region II,

At region III,

At region IV,

Electric Fields in Conductors Practice Problems