 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/5/d/ii/solution

For

κ > Ω

, put

η = κ^{2} - Ω^{2}

, so

0 < η < κ

and the poles are

i (κ \pm η)

. Closing in the upper half-plane for

t > 0

again includes both simple poles. The residue theorem gives

G (t) = e^{- κ t} \frac{sinh ( η t )}{η} = \frac{e ^{- (κ - η) t} - e ^{- (κ + η) t}}{2 η} .

This overdamped response has no oscillation. Its slow tail is

e^{- (κ - η) t} / (2 η)

, and it tends to zero because

η < κ

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