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differentiation help

Question from IsaacPhysics:
The displacement x of a damped, simple harmonic oscillator as a function of time t is given by x=Ae^(−αt) cos(ωt+ϕ). Find a general expression for the times at which the displacement is a maximum or minimum. (Your answer will involve an inverse trig function and an arbitrary integer n.)

My (wrong) answer: t = (arctan( - (alpha)/(omega)) - phi)/(omega)
I differentiated the expression for x with respect to t, and set dx/dt = 0 and rearranged for t.
I'm not sure why it's wrong or where to get the 'arbitrary integer n' from?
Reply 1
Looks about right though it would be handy to have the question link to check. The way youve written it is that there is a single solution (corresponding to arctan) whereas obviously there are muliptle solutions corresponding to the "+ n*pi".
Reply 2
Original post by mqb2766
Looks about right though it would be handy to have the question link to check. The way youve written it is that there is a single solution (corresponding to arctan) whereas obviously there are muliptle solutions corresponding to the "+ n*pi".

Thank you so much - got the answer now! :smile:
(part D of https://isaacphysics.org/questions/stationary_points4?board=d1084b81-6d7b-448f-a695-b76606b43f37&stage=a_level)

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