Critically damped solution to a damped driven oscillator

A damped, driven oscillator is described by the equation,

$$m\frac{d^2x}{dt^2}+b\frac{dx}{dt} + kx = F_0\cos(\omega t).$$

If $b^2=4km$, the system is critically damped and the solution has the form,

$$x(t) = C_1 \exp \left(-t/\tau\right)+C_2t \exp \left(-t/\tau\right)+\frac{F_0}{\rho}\cos (\omega t -\theta),$$


$$\rho = \sqrt{(k-m\omega^2)^2+\omega^2b^2},\qquad\tan\theta = \frac{\omega b}{k-m\omega^2},\qquad \tau=\frac{2m}{b},$$ $$C_1 = x_0-\frac{F_0}{\rho}\cos ( -\theta),\qquad C_2 = v_{x0} +\frac{C_1}{\tau} +\frac{\omega F_0}{\rho}\sin ( -\theta).$$

Here $x_0$ is the position at $t=0$ and $v_{x0}$ is the velocity at $t=0$.





$m=$ 1 [kg]

$F_0=$ 1 [N]

$b=\sqrt{4km}=$ 0.1 

$\omega=$ 1 [rad/s]

$k=$ 1 [N/m]

$x_0=$ 1 [m]

$v_{x0}=$ 1 [m/s]