Given a system differential equation it is possible to derive a pole-zero model directly, but it is more convenient to go first derive the transfer function, and then go from the transfer function to the pole-zero model.
Find a pole-zero model for the system described by the differential equation:
Step 1: Find the transfer function using the methods described here (1DE ↔ TF)
Step 2: Find a pole-zero representation using the methods described here (TF ↔ PZ). The pole-zero representation consists of:
Repeat the previous example, but reverse the order (i.e., do step 2 (PZ → TF) then step 1 (TF → DE).