 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/b/solution

Apply the Fourier transform of a derivative twice to the damped harmonic oscillator equation. Under the corresponding convergence assumptions,

(- ω^{2} + 2 iκω + Ω^{2}) y (ω) = f (ω) .

Thus

y (ω) = h (ω) f (ω), h (ω) = \frac{1}{Ω ^{2} - ω ^{2} + 2 iκω} .

Since

κ, Ω > 0

, this denominator has no real zero: its imaginary part could vanish only at

ω = 0

, where its real part is positive. The transform therefore defines a regular response on the real frequency axis.

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