Position of Cart attached to Wall by Spring under Damping/Overdamped

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:
 * $x = C_1 e^{m_1 t} + C_2 e^{m_2 t}$

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 $m^2 + 2 b + a^2 = 0$:
 * $m_1 = -b + \sqrt {b^2 - a^2}$
 * $m_2 = -b - \sqrt {b^2 - a^2}$

Such a system is defined as being overdamped.

Proof
When $b > a$, we have $b^2 - a^2 > 0$ and so $m_1$ and $m_2$ are real and distinct.

So from Solution of Constant Coefficient Homogeneous LSOODE: Real Roots of Auxiliary Equation:
 * $\mathbf x = C_1 e^{m_1 t} + C_2 e^{m_2 t}$

where