Question

Write down the differential equation for a damped harmonic oscillator

Answer 1

The Newton's second law for damped oscillator has a form

$$ma+cv+kx=0.$$

Thus, the equation of motion of a damped harmonic oscillator is as follows

$$m\frac{d^2x(t)}{dt^2}+c\frac{dx(t)}{dt}+kx(t)=0,$$

or

$$\ddot{x}(t)+2\beta\dot{x}(t)+\omega_0^2x(t)=0.$$

Here

$$2\beta=\frac{c}{m},\quad \omega_0^2=\frac{k}{m}.$$