Question

The differential equation of displacement of all "Simple harmonic motions" of given period \[\frac{{2\pi }}{{\rm{n}}}\], is
A) \[\frac{{{d^2}x}}{{d{t^2}}} + nx = 0\]
B) \[\frac{{{d^2}x}}{{d{t^2}}} + {n^2}x = 0\]
C) \[\frac{{{d^2}x}}{{d{t^2}}} - {n^2}x = 0\]
D) \[\frac{{{d^2}x}}{{d{t^2}}} + \frac{1}{{{n^2}}}x = 0\]

Answer 1

Hints:
In order to simplify the above expression, use the SHM standard differential equation. Find the oscillation frequency by comparing the two, and then utilize the relationship between frequency and time period to determine the time period.
Formula use:
 \[y = {x^n}\]
Differentiate with respect to x
\[\frac{{dy}}{{dx}} = n{x^{n-1}}\]
\[y = \cos (x)\]
Differentiate with respect to x
\[\frac{{dy}}{{dx}} = -\sin (x)\]

Complete step-by-step solution:
We have been given the differential equation of displacement of all "Simple harmonic motions" of given period as \[\frac{{2\pi }}{{\rm{n}}}\]
We have been given the displacement\[x\]for all SHM as
\[x = a\cos (nt + b)\]-- (1)
Differentiation is linear. We can differentiate summands separately and pull out constant factors.
Let’s solve the above equation (1) by first order differentiation with respect to t:\[ \Rightarrow \frac{{dx}}{{dt}} = - na\sin (nt + b)\]-- (2)
Apply the differentiation rule:
Solve the equation (2) by second order differentiation with respect to t:
\[ \Rightarrow \frac{{{d^2}x}}{{d{t^2}}} = - {n^2}a\cos (nt + b)\]
According to the equation (1), replace the value \[a\cos (nt + b)\] as \[x\]:
Since, \[x = a\cos (nt + b)\]
\[ \Rightarrow \frac{{{d^2}x}}{{d{t^2}}} = - {n^2}x\]
Let’s restructure the equation by explicitly having all the terms on one side:
\[ \Rightarrow \frac{{{d^2}x}}{{d{t^2}}} + {n^2}x = 0\]
Therefore, the differential equation of displacement of all "Simple harmonic motions" of given period \[\frac{{2\pi }}{{\rm{n}}}\], is \[\frac{{{d^2}x}}{{d{t^2}}} + {n^2}x = 0\]
Hence, the option B is correct.
Note:
Students often make mistakes in finding differential equation for simple harmonic motions. Because, it involves trigonometry functions and differentiation functions. The solutions of differential equations of simple harmonic motions are verified by substituting the x values in the above differential equation for the linear simple harmonic motion. We have to differentiate the functions carefully to get the required solution.