= Quadratically damped oscillator
{title2=$x\prime\prime+\epsilon|x\prime|x\prime+x=0$}
For initial amplitude one, the <method of multiple scales> gives $x\simeq A(\epsilon t)\sin t$ with $A_T=-4A^2/(3\pi)$, hence $A=1/(1+4\epsilon t/(3\pi))$. Positive damping causes algebraic decay. Negative damping yields a formal finite-time amplitude pole, but the weak-damping assumption fails before it because $|\epsilon|A$ becomes order one. A leading approximation has order-$|\epsilon|$ displacement error on fixed slow-time intervals away from that pole.
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