Question

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

Answer 1

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