© 2020 Elsevier LtdAsymptotic expansions are of great interest and significance in the study of canard explosions in singularly perturbed systems. Several classical methods have been developed to compute such expansions. However, for the non-generic case considered in this letter, those methods fail to do so. There only exists an estimation on the first non-zero term in the literature. Our aim is to propose a new approach to find the asymptotic expansions iteratively. Moreover, the exact value of the first non-zero term for the non-generic case is provided in terms of Airy function. The provided numerical results validate our analytical approximations.