= Solution
Set $T_f=(k_1S_0)^{-1}$, $\epsilon=E_0/S_0$, $K=(k_{-1}+k_2)/(k_1S_0)$ and $\lambda=k_2/(k_1S_0)$. On fast time $\tau_f=t/T_f$,
$$
s_{\tau_f}=\epsilon[-s+(s+K-\lambda)c],\qquad c_{\tau_f}=s-(s+K)c.
$$
For $\tau_f=O(1)$, leading <substrate> remains $s=1$. The initial complex buildup therefore solves $c'=1-(1+K)c$, $c(0)=0$, giving
$$
\boxed{s\simeq1,\qquad c\simeq\frac{1-e^{-(1+K)\tau_f}}{1+K}.}
$$
Time belongs inside the exponential. Its placement outside the exponential bracket in the printed expression is a typographical error; the derived form satisfies both the kinetic equation and zero initial complex <concentration>.
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