What is free vibration and forced vibration?

Hint: First of all we have to understand what free vibration is. From its name it suggests such a type of vibration which is independent and does not need any external force is known as free vibration. On the other hand forced vibration is the vibration or oscillation which requires external force to occur.

Complete step-by-step solution:
Free Vibration-
If a body is made to vibrate by disturbing it from its state of equilibrium and leaving to vibrate itself, the body vibrates with its own definite natural frequency. Such a vibration, which takes place only under the influence of its own elastic force is called natural or free vibration.
The equation of free vibration can be written as $\dfrac{{{d^2}x}}{{d{t^2}}} + {\omega ^2}x = 0$ where $\omega = \sqrt {\dfrac{a}{m}} = $ natural angular frequency.
Forced vibration-
The amplitude of oscillations for a damped oscillatory motion of a body goes on decreasing with time due to loss of energy to overcome the resistive forces. If an external periodic force is supplied to the system to make up for the losses, the amplitude of vibration does not decay with time and the body will vibrate regularly under such periodic force. This vibration of the body is called forced vibration. So, we have to apply external periodic force to maintain the oscillation.
The equation of forced vibration is
$\dfrac{{{d^2}x}}{{d{t^2}}} + 2k\dfrac{{dx}}{{dt}} + {\omega ^2}x = f{e^{ipt}}$ where $k = \dfrac{b}{{2m}} = $ damping constant, $\omega = \sqrt {\dfrac{a}{m}} = $ natural angular frequency, $f = \dfrac{{{F_0}}}{m}$, $p = $ cyclic frequency of periodic force and $x = $ instantaneous displacement from rest.

Note: The frequency of free vibration depends upon mass, shape and elastic properties of the body. Ideally free vibration of a body can never be realized as the amplitude gets on decreasing due to the viscous forces in nature like air and others and due to many other external forces.