= Slow oscillator damping by a relaxing auxiliary variable
{title2=$A(\tau)=\exp[-3\tau/2-3(1-e^{-\tau})]$}
For $x''+3\epsilon yx'+x=2(y')^2$ and $y'=\epsilon(1+x-y)$ with leading oscillation $x=A(\tau)\sin t$, $\tau=\epsilon t$, averaging the auxiliary equation gives $Y'=1-Y$. Cancellation of resonant forcing in the oscillator gives $2A'+3YA=0$. With $Y(0)=3$ and $A(0)=1$, $Y=1+2e^{-\tau}$ and the displayed amplitude follows. Bounded oscillatory corrections repair the instantaneous auxiliary equation; secular terms in a fixed-time expansion show why the slow scale is needed.
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