There are numerous cases in physics where the value of a quantity and changes in that quantity are related. For example, the speed of an object depends on its acceleration; the radioactivity of a sample depends on the amount of the sample present. Except in highly idealised settings, the analysis of these cases requires students to recognize, set up, and solve an ordinary differential equation (ODE). In many universities, ODEs are studied in mathematics before they are applied in physics. However, the aims of mathematicians and physicists can be very different. Mathematics modules tend to emphasise theoretical aspects of ODEs. In contrast, physics modules often emphasise modelling. This project is a multi-stage investigation that began by identifying the issues experienced by physics students during their study of ODEs before addressing them through the design and implementation of a set of tutorials. Having surveyed a cohort of physics students who completed a typical service module on ODEs, we found that many of them possessed a fragmented concept image of ODEs and insufficient instrumental understanding. Fifteen tutorials were designed to address these issues. Issues with instrumental understanding (primarily the manipulation of exponents and indefinite integration), and broadening the students’ concept image were the focuses. The effectiveness of the tutorials was measured using immediate pre/post-testing, delayed post-tests, and interviews with students. Together, these revealed significant gains in the understanding of ODEs as well as an appreciation of the guided-inquiry approach employed. Although some gaps in instrumental understanding were found to remain, students are hampered more by an incomplete concept image of ODEs. By studying the conceptual difficulties of physics students with ODEs and designing a guided teaching-learning sequence, we have been able to improve students’ conceptual understanding of ODEs without impacting negatively on their instrumental understanding.