Position of Cart attached to Wall by Spring under Damping

Problem Definition
Let:
 * $a^2 = \dfrac k m$
 * $2 b = \dfrac c m$

Then the horizontal position of $C$ at time $t$ can be expressed as:
 * $x = \begin{cases}

C_1 e^{m_1 t} + C_2 e^{m_1 t} & : b > a \\ & \\ C_1 e^{-a t} + C_2 t e^{-a t} & : b = a \\ & \\ e^{-b t} \left({C_1 \cos \alpha t + C_2 \sin \alpha t}\right) & : b < a \end{cases}$

where:
 * $C_1$ and $C_2$ depend upon the conditions of $C$ at time $t = 0$


 * $m_1$ and $m_2$ are the roots of the auxiliary equation $m^2 + 2 b + a^2 = 0$:
 * $m_1 = -b + \sqrt {b^2 - a^2}$
 * $m_2 = -b - \sqrt {b^2 - a^2}$


 * $\alpha = \sqrt {a^2 - b^2}$