5.5 Forced oscillations and resonance
For the inhomogeneous system \(x' = Ax + g(t)\), variation of parameters (Theorem 4.3.1) becomes \[ x(t) = e^{A(t - t_0)}x_0 + \int _{t_0}^te^{A(t - s)}g(s)\,ds , \] a convolution of the forcing with the exponential. For forcing of the form \(e^{\mu t}\) times a polynomial, the integral can be done in closed form, which is the justification for the method of undetermined coefficients. The most important case is periodic forcing of an oscillator.
Example 5.5.1. The undamped oscillator \(y'' + \omega _0^2y = F\cos \omega t\) has, for \(\omega \neq \omega _0\), the particular solution \[ y_p = \frac {F}{\omega _0^2 - \omega ^2}\cos \omega t , \] whose amplitude grows without bound as \(\omega \to \omega _0\). At \(\omega = \omega _0\) the forcing frequency is an eigenvalue \(\pm i\omega _0\) of the system, and the particular solution is \(y_p = \frac {F}{2\omega _0}t\sin \omega _0t\), which grows linearly: resonance. With \(y(0) = y'(0) = 0\) and \(\omega \) close to \(\omega _0\), the solution is \[ y = \frac {F}{\omega _0^2 - \omega ^2}(\cos \omega t - \cos \omega _0t) = \frac {2F}{\omega _0^2 - \omega ^2}\sin \frac {(\omega _0 - \omega )t}{2}\sin \frac {(\omega _0 + \omega )t}{2} , \] a fast oscillation at the mean frequency modulated by a slow envelope, the phenomenon of beats. With damping \(2\zeta \omega _0y'\), the response amplitude \(F/\sqrt {(\omega _0^2 - \omega ^2)^2 + 4\zeta ^2\omega _0^2\omega ^2}\) stays finite and peaks near \(\omega _0\) when \(\zeta \) is small.
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