On the Instability and Critical Damping Conditions, $k\tau = 1/e$ and $k\tau = \pi/2$ of the equation $\dot{\theta} = -k \theta(t-\tau)$

Abstract

In this note, I show that it is possible to use elementary mathematics, instead of the machinery of Lambert function, Laplace Transform, or numerics, to derive the instability condition, $k \tau = \pi/2$, and the critical damping condition, $k\tau = 1/e$, for the time-delayed equation $\dot{\theta} = -k \theta(t-\tau)$. I hope it will be useful for the new comers to this equation, and perhaps even to the experts if this is a simpler method compared to other versions.