\begin{definition}[Definition:Critically Damped]
Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form:
:$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$
for $a, b \in \R_{>0}$.
Let $b = a$, so that the solution of $(1)$ is in the form:
:$x = C_1 e^{-a t} + C_2 t e^{-a t}$
Then $S$ is described as being '''critically damped'''.
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\end{definition}