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Write $T=\epsilon t$ and $q(T)=2e^{-T}$. The forcing acts on the fast initial-time scale, so first remove a localized particular solution. Expanding the coefficient there givesA particular choice is $p_1=(t+1/2)e^{-2t}$, henceIts initial value is $1+\epsilon /2$ and its initial derivative is $-2+O({\epsilon }^2)$. The localized forcing determines WKB phase constants; omitting it before imposing the initial conditions would miss the permanent free oscillation.

For the homogeneous part, the slow-time equation is ${\epsilon }^2{u}_{TT}+q^{2}u=0$. Write its WKB approximation as $A(T)[C\cos \Phi +B\sin \Phi ]$, normalizing $A(0)=1$. The leading amplitude transport equation gives $A=e^{T/2}$ and leading phase ${\epsilon }^{-1}{\int }_0^{T}q(s)ds=2(1-e^{-T})/\epsilon$.

To include every $O(\epsilon )$ contribution on bounded slow-time intervals, the first phase correction is needed. If the local phase wavenumber is $q+{\epsilon }^2{q}_2$, the real part of the amplitude-phase equation gives $q_2=A_{TT}/(2qA)=e^T/16$. The WKB phase correction for a slowly varying oscillator, anchored to zero at the initial time, is thereforeThere is no first-order bulk amplitude correction in this normalization. At $t=0$, the homogeneous initial value must be $-\epsilon /2+O({\epsilon }^2)$ and its initial derivative $1+O({\epsilon }^2)$. Since $A_t^{\prime }(0)=\epsilon /2$ and ${\Phi }_t^{\prime }(0)=2+O({\epsilon }^2)$, these give $C=-\epsilon /2+O({\epsilon }^2)$ and $B=1/2+O({\epsilon }^2)$. The requested forced approximation isfor $0\le \epsilon t\le T_0$ with fixed $T_0$. The approximate initial value is exactly one through this order and its derivative is $-1+O({\epsilon }^2)$. The particular-solution residual is uniformly localized and $O({\epsilon }^2)$; the corrected homogeneous residual is $O({\epsilon }^3)$ on the long interval, producing $O({\epsilon }^2)$ solution error after integration over times of order ${\epsilon }^{-1}$.

The WKB approximation requires the relative frequency-change parameterIt therefore fails when $\epsilon e^{\epsilon t}=O(1)$, even though the coefficient has no finite-time zero. This is the Bessel transition for an exponentially decaying oscillator. Introduce the shifted slow time and amplitudeThe exact rescaled equation isFor fixed $\tau$ its right side is smaller than every algebraic power of $\epsilon$. The leading transition problem is consequentlyWith $Z=2e^{-\tau }$ this becomes $Z^2{U}_{0,ZZ}+Z{U}_{0,Z}+Z^2{U}_0=0$, the order-zero Bessel differential equation. This identifies the governing equation without solving it.

The amplitude-phase series also supplies overlap data while $\epsilon e^T\ll 1$, although its bounded-$T$ absolute error estimate is not uniform up to the transition. For matching, combine the earlier sine and cosine terms into $\frac{1}{2}\sin (\Phi -\epsilon )$ through first order. In the overlap $\epsilon \ll e^{\tau }\ll 1$, their phase isto the retained order. Thus amplitude and phase matching as $\tau \to -\infty$ requireThese are oscillatory amplitude-phase conditions, rather than pointwise ratio statements at zeros. The smaller $e^{\tau }/16$ phase term records the first overlap correction if required. The rapidly varying matching constant ${\alpha }_{\epsilon }$ must be retained as data; there is no single epsilon-independent limiting phase. The two matching conditions determine the leading transition solution, which is deliberately left unsolved.