= Localized forcing determines WKB phase constants
{title2=$u=u_p+u_h$}
For a slowly varying oscillator with rapidly decaying forcing, first construct the localized particular solution $u_p$. Apply the <initial conditions> to the residual $u_h$ to determine its <WKB approximation> amplitude and phase constants. Ignoring the early forcing because it later becomes small loses the permanent free oscillations it excites. In $u''+4e^{-2\varepsilon t}u=8e^{-2t}$, the particular expansion is $u_p=e^{-2t}[1+\varepsilon(t+1/2)]+O(\varepsilon^2)$, whose initial value shifts the first homogeneous cosine coefficient by $-\varepsilon/2$.
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