= Solution
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,
$$
x_1''+x_1=-9\cos t,\qquad y_1'=\sin t-2,
$$
with zero initial corrections. Therefore
$$
x_1=-\frac92t\sin t,\qquad y_1=1-\cos t-2t.
$$
The secular terms become comparable with the leading terms at $t=O(\epsilon^{-1})$. \b[The fixed-time expansion is not uniform on the slow relaxation time.]
Introduce $\tau=\epsilon t$ through the <method of multiple scales>, and write $x=A(\tau)\sin t+O(\epsilon)$, $y=Y(\tau)+O(\epsilon)$. Average the second equation over the rapid oscillation. Since the sine has zero mean, $Y'=1-Y$, where the prime now denotes $d/d\tau$. With $Y(0)=3$, $Y=1+2e^{-\tau}$.
In the oscillator equation, $2(y')^2$ is of order $\epsilon^2$. The order-$\epsilon$ resonant cosine forcing is $-(2A'+3YA)\cos t$. A bounded first correction requires its coefficient to vanish, so
$$
2A'+3YA=0,\qquad A(0)=1.
$$
Integration gives the <slow oscillator damping by a relaxing auxiliary variable>:
$$
\boxed{A(\tau)=\exp\left[-\frac32\tau-3(1-e^{-\tau})\right].}
$$
Hence the uniform leading solution is
$$
\boxed{x(t)=e^{-3\epsilon t/2-3(1-e^{-\epsilon t})}\sin t+O(\epsilon),\qquad y(t)=1+2e^{-\epsilon t}+O(\epsilon).}
$$
To check the uniform ordering, the instantaneous sine forcing in the auxiliary equation is absorbed by a bounded first correction $-\epsilon A(\tau)\cos t$, plus a bounded slow correction chosen to match the initial data. The resonant first-order oscillator forcing has already been removed. Remaining order-$\epsilon^2$ terms accumulate by at most $O(\epsilon)$ over $0\leq t\leq K/\epsilon$ 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.
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