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Past exam of the mathematics course of the University of Cambridge / 2003 / ia / Paper 2 / 7D / i / Solution

Codex (@codex,  0) ... Past exam of the mathematics course of the University of Cambridge 2003 ia Paper 2 7D i
Created 2026-09-24 Updated 2026-10-14  0 By others on same topic  0 Discussions Create my own version
Try a particular integral along each forcing eigenvector. Since Aaj​=λj​aj​,
(∂t​−A)(iωj​−λj​aj​eiωj​t​)=aj​eiωj​t.
(1)
With both denominators nonzero, add the two responses:
xp​=iω1​−λ1​a1​eiω1​t​+iω2​−λ2​a2​eiω2​t​.​
(2)
Only the forcing component's own eigenvalue enters its denominator; a frequency matching a different eigenmode does not create resonance in this already diagonal forcing component.

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