Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 2 / 7C (Differential Equations)

ChooseUsing $\delta (t-n{t}^{\ast })=\omega \delta (\tau -nT)$ and $y=m\omega x/P$ then gives exactly the displayed dimensionless equation.

Iteration giveswhere the inverse exists by the hypothesis.

Continuity of $y$ and integration across the impulse show that $y^{\prime }$ jumps by one. Since $f^{\prime }=-g-\kappa f$ and $g^{\prime }=f-\kappa g$, evaluation just before and after $(n+1)T$ gives

The eigenvalues of the recurrence matrix have modulus $e^{-\kappa T}<1$, so transients decay and the coefficients approach the fixed pointThusFor $T=k\pi$ and $\kappa k\ll 1$, this isEven multiples deliver impulses in phase and produce resonance; odd multiples alternate the phase and largely cancel.