Consider the singularly perturbed system of ordinary differential equations \(\epsilon\) \({\dot \xi}=F(\xi,\eta,\epsilon)\), \({\dot \eta}=G(\xi,\eta,\epsilon)\) with \(\xi,F\in R^{\nu}\), \(\eta,G\in R^{\mu}\), \((\xi,\eta)\in \Omega \subset R^{\nu +\mu}\), \(\epsilon \in [0,\epsilon_ 0)\), \(F,G\in C^{r+2}(\Omega \times [0,\epsilon_ 0))\), \(r\geq 0\). The solution of this system can be written as \[ \left( \begin{matrix} \xi (t,\epsilon)\\ \eta (t,\epsilon)\end{matrix} \right)=\Gamma (t/\epsilon,\epsilon)+\gamma (t,\epsilon) \] where \(\Gamma\),\(\gamma\) ae \(C^{r+1}\) in their arguments and \(| \Gamma (\tau,\epsilon)| \leq C\cdot e^{-\delta \tau};\) \(\Gamma\) (\(\cdot,\cdot)\) is called ``inner solution'' and \(\gamma\) (\(\cdot,\cdot)\) is called ``outer solution''. It is shown that an outer solution has a certain asymptotic expansion with respect to \(\epsilon\) ; a method of computing the coefficients of the expansion is given. The results are applied to a model of enzyme reaction system.