Reduce a coupled fast–slow subsystem without losing its startup state

Consider the nondimensional coupled engineering model \[ x'=-x+y, \qquad \varepsilon y'=x-2y, \qquad x(0)=1, \qquad y(0)=0, \] with $0<\varepsilon\ll1$. Here $x$ is a slow stored quantity and $y$ is a fast auxiliary subsystem. Set $\varepsilon=0$ to derive the slow manifold and the reduced IVP for $x$. Solve for the leading outer approximation to both variables. On the fast time $\tau=t/\varepsilon$, freeze the leading slow state and solve the inner problem needed to satisfy $y(0)=0$. Form a leading composite approximation for $y$ and explain why no order-one initial layer is needed in $x$. Independently compute the two exact eigenvalues of the full coefficient matrix. Show which is slow and which is fast by expanding them for small $\varepsilon$. Write $x=C_s e^{\lambda_s t}+C_f e^{\lambda_f t}$, determine $C_s,C_f$ from the initial data, and show that $C_f=O(\varepsilon)$ even though the fast modal contribution to $y=x'+x$ is $O(1)$. For $\varepsilon=0.0200$, compute both eigenvalues. Explain when eliminating $y$ is valid and what startup information the algebraic relation $y=x/2$ discards.

The reduced relation is $y=x/2$, giving $x'=-x/2$.