A 2 kg mass is attached to a spring with a spring constant $k = 4 \, N/m$ and a damping coefficient of $b = 5 \, kg/s$. This mass is also subjected to an external sinusoidal force given by $F_0 \cos(\omega t)$, where the amplitude $F_0 = 10 \, N$ and the frequency $\omega = 2 \, rad/s$. The motion of the mass, described by the displacement $x(t)$ over time, follows the differential equation:
$$m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = F_0 \cos(\omega t)$$
Given the initial conditions $x(0) = 0$ and $\frac{dx}{dt}(0) = 0$, determine the expression for $x(t)$ and identify the steady-state solution, which describes the long-term oscillatory behavior of the system.
