 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2015/11/ii/solution

After the force is removed, the damped harmonic oscillator evolves freely. There is no impulsive force at zero, so both displacement and velocity are continuous. Matching the values in part (i) gives

x (t) = e^{- γ t} [x_{0} cos ν t + \frac{v _{0} + γ x _{0}}{ν} sin ν t], t > 0.

Equivalently,

x (t) = \frac{A ω e ^{- γ t}}{N} [- 2 γ cos ν t + \frac{Ω ^{2} - ω ^{2} - 2 γ ^{2}}{ν} sin ν t] .

This is bounded in positive time, completing the unique globally bounded solution. The formulas also include

ω = 0

, for which the forcing and solution both vanish.

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