Which of the following is the valid differential equation x = a cos(αt + β)?

Category: QuestionsWhich of the following is the valid differential equation x = a cos(αt + β)?
Editor">Editor Staff asked 11 months ago

Which of the following is the valid differential equation x = a cos(αt + β)?
 
(a) d^2x/dt^2 – αx = 0
 
(b) d^2x/dt^2 + αx = 0
 
(c) d^2x/dt^2 – α^2x = 0
 
(d) d^2x/dt^2 + α^2x = 0
 
This question was addressed to me in a national level competition.
 
This intriguing question originated from Linear Second Order Differential Equations topic in chapter Differential Equations of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
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1 Answers
Editor">Editor Staff answered 11 months ago

Right option is (d) d^2x/dt^2 + α^2x = 0
 
For explanation I would say: Since, x = a cos(αt + β)
 
Therefore, dx/dt = a cos(αt + β)
 
And, d^2x/dt^2 = -a α^2 cos(αt + β)
 
= -α^2 a cos(αt + β)
 
Or, d^2x/dt^2 = -α^2 [as a cos(αt + β) = x]
 
So, d^2x/dt^2 + α^2x = 0