This study investigated six students' understandings of and difficulties with qualitative and numerical methods for analyzing differential equations. From an individual cognitive perspective, the obstacles found to influence the development of students' understandings included the function dilemma, the tendency to overgeneralize, interference from informal or intuitive notions, and the complexity involved with graphical interpretations. From a sociocultural perspective, students' understandings were constrained by the use of technology that was disconnected from the learning process, instruction that did not seek out students' explanations, and classroom interactions that implicitly established procedure-based mathematical justifications. Contains 32 references. (Author/NB)