It is recently discussed that a plasma system with drifting ions at near the acoustic phase speed becomes susceptible to weak but finite electron inertial delay effect. In such plasma condition, the linear normal modes of acoustic oscillations undergo resonant linear growth. Such situation exists naturally in transonic zones of boundary plasma layers. The present contribution considers this as a specific case to describe the possible role of an external source driving on the usual ion acoustic soliton solution in the unstable region of the transonic zone assumed to have finite extension. A driven Korteweg–de Vries equation is obtained that is not exactly integrable. It is shown by numerical analysis that the usual acoustic soliton solution exists only for infinitely long wavelength source perturbation. For short wavelength (but within the limit of the long wavelength approximation) source perturbations interesting solutions of single unit of localized oscillatory shock-like structure are obtained. Possible physical interpretations of the results are included in the text.