This paper develops an algorithm which, given an input-output differential equation, produces under appropriate circumstances a parametrization of the observation space of the underlying state-space system. The paper extends previous results by the authors to a more general setting, so that state-space systems, in which the controls enter nonlinearly, can be accommodated and, most importantly, the paper deals with implicitly defined systems at both the input-output and state-space level. Fliess and Hasler have demonstrated the importance of implicit input-output models, which yield only local state-space descriptions. Various realizations are obtained, depending on the properties of the derived candidate observation space. Examples of the algorithm are described, with the aid of an implementation using the computer algebra system AXIOM.