
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
163.8k+ views
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.
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.
Recently Updated Pages
Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

Atomic Structure - Electrons, Protons, Neutrons and Atomic Models

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Learn About Angle Of Deviation In Prism: JEE Main Physics 2025

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths In Hindi Chapter 1 Sets

Instantaneous Velocity - Formula based Examples for JEE

NCERT Solutions for Class 11 Maths Chapter 8 Sequences and Series
