PX153 - B5.2 - overdamping

x (t) = e^{- α t} (C e^{κ t} + D e^{- κ t})

there is no oscillation
for the initial conditions: $x (0) = x_{0,} v (0) = 0$
$x (0) = C + D = x_{0}$
$(κ - α) C - (κ + α) D = 0 ⟹ D = \frac{κ - α}{2 κ} x_{0}, C = \frac{κ + α}{2 κ} x_{0}$

∴ x (t) = \frac{x_{0}}{2 κ} e^{- α t} ((κ + α) e^{κ t} + (κ - α) e^{- κ t})

for long times: $e^{κ t} >> e^{- κ t}$ $⟹ x (t) \to e^{- α t} \cdot e^{κ t} = e^{- (α - κ) t}$
- ie: exponential decay with time constant longer than $\frac{1}{α}$