Obtain the differential equation of forced oscillation.

(N/A) To sustain oscillations in a system,an external periodic force is applied,given by:
$F(t) = F_{0} \cos \omega_{d} t$
where $F_{0}$ is the amplitude and $\omega_{d}$ is the angular frequency of the driving force.
Three forces act on the oscillator:
$(1)$ Restoring force: $F_{r} = -k x(t)$
$(2)$ Resistive (damping) force: $F_{s} = -b v(t) = -b \frac{dx}{dt}$
$(3)$ External periodic force: $F_{d} = F_{0} \cos \omega_{d} t$
According to Newton's second law of motion,the net force is $F_{net} = ma(t) = m \frac{d^{2}x}{dt^{2}}$.
Thus,$m \frac{d^{2}x}{dt^{2}} = F_{r} + F_{s} + F_{d}$
$m \frac{d^{2}x}{dt^{2}} = -kx - b \frac{dx}{dt} + F_{0} \cos \omega_{d} t$
Rearranging the terms,we get the differential equation of forced oscillation:
$m \frac{d^{2}x}{dt^{2}} + b \frac{dx}{dt} + kx = F_{0} \cos \omega_{d} t$
Dividing by mass $m$,we obtain:
$\frac{d^{2}x}{dt^{2}} + \frac{b}{m} \frac{dx}{dt} + \frac{k}{m} x = \frac{F_{0}}{m} \cos \omega_{d} t$