Resolve the startup layer hidden by a quasi-steady sensor model

A fast first-order sensor with time constant $\varepsilon>0$ seconds is exposed at $t=0$ to a unit step in a nondimensionalized measured quantity. Its output satisfies \[ \varepsilon y'(t)+y(t)=1, \qquad y(0)=0, \qquad 0\le t\le T, \] where $T>0$ is fixed independently of $\varepsilon$. Solve the initial-value problem exactly. Derive the outer reduced model as $\varepsilon\to0$, identify why it cannot satisfy the initial condition, and introduce the stretched time $\tau=t/\varepsilon$ to derive the leading inner problem. Construct the standard leading composite approximation and determine whether it happens to equal the exact solution here. Prove that $y_\varepsilon(t)\to1$ for every fixed $t>0$ but not uniformly on $[0,T]$. Show that convergence is uniform on every $[\delta,T]$ with $\delta>0$. Derive the time needed to enter and remain within one percent of the quasi-steady value, and compute the exact area $\int_0^T|1-y_\varepsilon(t)|\,dt$. Evaluate the one-percent settling time for $\varepsilon=0.0200\ \mathrm s$. Explain why simply setting $\varepsilon=0$ is reliable after startup but gives a false initial value.

The exact homogeneous factor is $e^{-t/\varepsilon}$.