Given a system differential equation it is possible to derive a state space model directly, but it is more convenient to go first derive the transfer function, and then go from the transfer function to the state space model.
Find a state space 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 state space representation using the methods described here (TF ↔ SS). In this case we are using a CCF form).
Repeat the previous example, but reverse the order (i.e., do step 2 (SS → TF) then step 1 (TF → SS)).