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

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

Then the horizontal position of $C$ at time $t$ is determined by the roots of the auxiliary equation:
 * $m^2 + 2 b + a^2 = 0$

according to Solution of Constant Coefficient Homogeneous LSOODE.

Proof
From Motion of Cart attached to Wall by Spring under Damping, the horizontal position of $C$ is given as:
 * $\dfrac {\mathrm d^2 \mathbf x} {\mathrm d t^2} + \dfrac c m \dfrac {\mathrm d \mathbf x} {\mathrm d t} + \dfrac k m \mathbf x = 0$

With the given substitutions $a$ and $b$, this resolves to:


 * $\dfrac {\mathrm d^2 \mathbf x} {\mathrm d t^2} + 2 b \dfrac {\mathrm d \mathbf x} {\mathrm d t} + a^2 \mathbf x = 0$

This is a homogeneous linear second order ODE with constant coefficients.

Recall that $m_1$ and $m_2$ are the roots of the auxiliary equation:
 * $m^2 + 2 b + a^2 = 0$

By Solution to Quadratic Equation: Real Coefficients:

From the initial problem definition, we have that $k, m, c \in \R_{>0}$.

Hence $a, b \in \R_{>0}$.

Hence the nature of $m_1$ and $m_2$ is dependent upon whether $b > a$, $b = a$ or $b < a$.