= Exact late slope of an exponentially damped oscillator
{title2=$\lim_{t\to\infty}y_t=e^{-1/2}M(-1/(2\varepsilon^2),1,1/2)$}
For $y_{tt}+e^{-2\varepsilon t}(\varepsilon y_t+y)=0$, $y(0)=0$, $y_t(0)=1$, set $r=e^{-2\varepsilon t}/2$. The exact equation becomes $ry_{rr}+(1-r)y_r+y/(2\varepsilon^2)=0$. Let $m(t)=M(-1/(2\varepsilon^2),1,r)$, using the <Kummer function>. It is regular at $r=0$ and tends to one. The <Wronskian> $W=my_t-m_ty$ obeys $W_t=-\varepsilon e^{-2\varepsilon t}W$, with $W(0)=M(-1/(2\varepsilon^2),1,1/2)$. Integration gives the displayed exact late slope. The other local solution grows at most logarithmically in $r$, so $m_ty\to0$ and $W\to y_t$. This provides a check on a <WKB approximation> even near phases where its leading predicted slope vanishes.
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