Establish the equation of motion of a weakly damped forced oscillator explaining the significance of each term. Differentiate between transient and steady state of the oscillator.

The equation of motion for a weakly damped forced oscillator can be derived from the general equation of motion for a damped harmonic oscillator subjected to an external driving force. The equation is expressed as:

\[
m \ddot{x} + b \dot{x} + kx = F_0 \cos(\omega t)
\]

Where:
- \(m\) is the mass of the object attached to the oscillator.
- \(x(t)\) is the displacement of the object from the equilibrium position as a function of time.
- \(\ddot{x}\) is the acceleration of the object (second derivative of displacement).
- \(b\) is the damping coefficient that characterizes the resistive force opposing the motion.
- \(\dot{x}\) is the velocity of the object (first derivative of displacement).
- \(k\) is the spring constant, determining the restoring force acting on the oscillator.
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