PX153 - B5.3 - critical damping

$α^{2} = ω_{0}^{2}$
guess solution: $x (t) = A e^{λ t}$
$λ^{2} + 2 α λ + ω_{0}^{2} = 0$
$λ_{1} = - α + \sqrt{α^{2} - ω_{0}^{2}}, λ_{2} = - α + \sqrt{α^{2} - ω_{0}^{2}}$
only one solution: $λ = - α$
must have two constants of integration for the second order differential equation, therefore there must be another term
postulate a second solution:

x (t) = D t e^{λ t} = D t e^{- α t}

\frac{d x}{d t} = D e^{- α t} (1 - α t)

\frac{d^{2} x}{d t^{2}} = α D e^{- α t} (α t - 2)

α (α t - 2) D e^{α t} + 2 α (1 - α t) D e^{- α t} + ω_{0}^{2} t D e^{- α t} = 0

α^{2} t - 2 α + 2 α - 2 α^{2} t + ω_{0}^{2} t = 0

(ω_{0}^{2} - α^{2}) t = 0

for the initial conditions $x (0) = x_{0}$ , $v (0) = 0$ :
$x_{0} = C$
$- α C + D = 0$

∴ x (t) = x_{0} (e^{- α t} + α t e^{- α t})