= Order reduction of a radiation-reaction oscillator
For $|\varepsilon\omega|\ll1$, use the zeroth-order equation $\ddot x=f-x$ inside the small third-derivative term, giving
$$
\ddot x+\varepsilon\dot x+x=f+\varepsilon\dot f.
$$
For positive $\varepsilon$ this reduced equation has a stable retarded response with no high-frequency runaway. Its response $G_{\rm red}=(1-i\varepsilon\omega)/(1-\omega^2-i\varepsilon\omega)$ agrees with the unreduced oscillator to first order at fixed nonresonant frequency. Order reduction deliberately removes the singular fast mode outside the assumed low-frequency regime; it is an approximation rather than an exact alternative boundary condition for the third-order equation.
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