Second-order differential equations

Introduction

We consider the general Second-order differential equation

$\tau^2 \frac{d^2y(t)}{d t^2} + 2 \zeta \tau \frac{d y(t)}{d t} + y(t)= x(t)$

If you expand the previous Second-order differential equation:

$\tau_1 \tau_2 \frac{d^2y(t)}{d t^2} + ( \tau_1 + \tau_2 ) \frac{d y(t)}{d t} + y(t) = x(t)$

Where:

$\tau = \sqrt{\tau_1 \tau_2}$

$\zeta = \frac{\tau_1 + \tau_2 }{2 \sqrt{\tau_1 \tau_2}}$