When an external force with angular frequency $\omega_d$ acts on a system of natural angular frequency $\omega$,the system oscillates with angular frequency $\omega_d$. The condition for the amplitude of oscillations to be maximum is

In forced oscillations,a particle oscillates simple harmonically with a frequency equal to

$A$ particle with a restoring force proportional to displacement and a resisting force proportional to velocity is subjected to a driving force $F \sin \omega t$. If the amplitude of the particle is maximum for $\omega = \omega_1$ and the energy of the particle is maximum for $\omega = \omega_2$,then (where $\omega_0$ is the natural frequency of oscillation of the particle):

The graph between velocity and position for a damped oscillation will be:

The amplitude of a damped oscillator becomes $\left(\frac{1}{3}\right)$ of its original amplitude in $2 \ s$. If its amplitude after $6 \ s$ becomes $\left(\frac{1}{n}\right)$ times the original amplitude,the value of $n$ is ($n$ is a non-zero integer).

Derive the differential equation for damped oscillations and write its solution.