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A singular perturbation is one for which setting the parameter to zero does not give a regular approximation uniformly over the region of interest. The limiting equation may lose a derivative or boundary condition, or a formally small correction may accumulate over a long time. Here the last mechanism occurs.

At fixed $t$, substitute a regular asymptotic expansion. The leading equations and initial conditions give $y_0=3$ and $x_0=\sin t$. At first order,with zero initial corrections. ThereforeThe secular terms become comparable with the leading terms at $t=O({ϵ}^{-1})$. The fixed-time expansion is not uniform on the slow relaxation time.

Introduce $\tau =ϵt$ through the method of multiple scales, and write $x=A(\tau )\sin t+O(ϵ)$, $y=Y(\tau )+O(ϵ)$. Average the second equation over the rapid oscillation. Since the sine has zero mean, $Y^{\prime }=1-Y$, where the prime now denotes $d/d\tau$. With $Y(0)=3$, $Y=1+2e^{-\tau }$.

In the oscillator equation, $2(y^{\prime }{)}^2$ is of order ${ϵ}^2$. The order-$ϵ$ resonant cosine forcing is $-(2A^{\prime }+3YA)\cos t$. A bounded first correction requires its coefficient to vanish, soIntegration gives the slow oscillator damping by a relaxing auxiliary variable:Hence the uniform leading solution isTo check the uniform ordering, the instantaneous sine forcing in the auxiliary equation is absorbed by a bounded first correction $-ϵA(\tau )\cos t$, plus a bounded slow correction chosen to match the initial data. The resonant first-order oscillator forcing has already been removed. Remaining order-${ϵ}^2$ terms accumulate by at most $O(ϵ)$ over $0\le t\le K/ϵ$ for fixed $K$. The slow coefficients stay bounded on this interval, and variation of constants for the oscillator and the auxiliary first-order equation gives the displayed uniform errors. Expanding the amplitude near $\tau =0$ gives $A=1-9\tau /2+\cdots$, recovering the secular term of the failed regular expansion.