Overdamping

For $\zeta >1$, let $r_{+}>r_{-}$ be the real characteristic roots. The response with the specified initial conditions isThe generic late response is governed by $r_{+}$, the root nearer zero. For very large damping ratio, $r_{+}\sim -{\omega }_0/(2\zeta )=-k/b$: an increasingly strong brake delays the return to equilibrium. This approximation comes from balancing the spring and viscous damping terms after the fast transient. Choosing $v_0=r_{-}{x}_0$ makes $C_{+}=0$ and leaves only the faster root; that exception requires different initial velocities as $b$ varies.