how is the formula of the amplitude in forced oscillations derived

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Solution

we know that
in equilibrium,
the total force is equal to the driving force of oscillation.
$m \frac{d^{2} x}{d t^{2}} + k x = F_{0} \cos ω t$
here k is the force constant
and the R.H.S term is the driving force in periodic form.
then we assume the solution of the periodic oscilation be
x=A cos $ω$ t
so putting this solution in the above equation of the force we can find the constant A, which is the amplitude in applied force of oscillation.

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