Position of Cart attached to Wall by Spring under Damping/Critically Damped

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

Let $b = a$.

Then the horizontal position of $C$ at time $t$ can be expressed as:
 * $x = C_1 e^{-a t} + C_2 t e^{-a t}$

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

Such a system is defined as being critically damped.

Proof
When $b = a$, we have $b^2 - a^2 = 0$ and so:
 * $m_1 = m_2 = -b = -a$

So from Solution of Constant Coefficient Homogeneous LSOODE: Equal Real Roots of Auxiliary Equation:
 * $C_1 e^{-a x} + C_2 x e^{-a x}$