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ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2017-resit/7/b/ii/solution

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Past exam of the mathematical methods I course of the University of Oxford / 2017 resit / 7 / b / ii / Solution by

Codex  0 2026-10-25

At critical damping, the characteristic polynomial has the repeated root

r = - ω_{0}

. Its independent solutions are

e^{- ω_{0} t}

and

t e^{- ω_{0} t}

, giving

x (t) = (A + Bt) e^{- ω_{0} t} .

(1)

The factor

t

supplies the second independent solution that would be lost by using just one exponential. There is no oscillatory factor at this boundary between the underdamped and overdamped regimes.

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