A slightly different system

A damped oscillator is modeled as a mass $m$ at equilibrium point $x = x_{\rm eq}$ acted on by (1) an ideal spring of spring constant $k$ and (2) a viscous drag force proportional to the velocity: $\vec{F}_{\rm drag} = -bm\vec{v}$.

(a)

Derive the equation of motion.
Hint: You can check (especially your signs) against Eq. (13-41) of YF, p. 411. Be careful in comparing, though, because their ``$b$'' is actually our ``$b \cdot m$'' and thus your equation won't look exactly like theirs.

(b)

Verify that $x(t) = x_{\rm eq} + Ae^{-bt/2}\sin(\omega_1 t + \delta_o)$ is a general solution of the equation of motion. What are the adjustable parameters? What value must $\omega_1$ have in terms of $\omega_o \equiv \sqrt{k/m}$ and $b$?

Hint: Keep your work organized, and first show

$$\dot x =-\frac{b}{2} \left(x-x_{\rm eq}\right) + \omega_1 A e^{-bt/2} \cos(\omega_1 t + \delta_o),$$

and then show

$$\ddot{x} = \left(\frac{b^2}{4}-\omega_1^2\right)\left(x-x_{\rm eq}\right)-b \omega_1 A e^{-bt/2} \cos(\omega_1 t + \delta_o),$$

before substituting into the equation of motion.