Localized forcing determines WKB phase constants ($u=u_p+u_h$)

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 $u^{\prime \prime }+4e^{-2\epsilon t}u=8e^{-2t}$, the particular expansion is $u_p=e^{-2t}[1+\epsilon (t+1/2)]+O({\epsilon }^2)$, whose initial value shifts the first homogeneous cosine coefficient by $-\epsilon /2$.