An undamped oscillator obeys

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Simple, damped and driven oscillations

Mathematical Methods for Physics, Ch Higher-order linear equations, the Wronskian and variation of parameters

Problem

An undamped oscillator obeys x¨+ω02x=s(t)\ddot{x} + \omega_0^2 x = s(t), where s(t)=Fsec⁡(ω0t)tan⁡(ω0t)s(t) = F\sec(\omega_0 t)\tan(\omega_0 t) for ∣ω0t∣<π/2|\omega_0 t| < \pi/2. Here FF is a constant with units of acceleration and ω0\omega_0 is the natural angular frequency of the undriven oscillator. The oscillator starts from rest at its equilibrium position, x(0)=0x(0) = 0, x˙(0)=0\dot{x}(0) = 0. ((a)) State why the method of undetermined coefficients cannot supply a particular integral for this drive. ((b)) Use variation of parameters, with y1=cos⁡ω0ty_1 = \cos\omega_0 t and y2=sin⁡ω0ty_2 = \sin\omega_0 t, to find x(t)x(t) on ∣ω0t∣<π/2|\omega_0 t| < \pi/2. ((c)) Evaluate xx at t=1.0 st = 1.0\,\mathrm{s} for ω0=1.0 rad s−1\omega_0 = 1.0\,\mathrm{rad\,s^{-1}} and F=1.0 m s−2F = 1.0\,\mathrm{m\,s^{-2}}.

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