Methods of teaching differential equations (DE) have long focused on rote analytic solution methods that align poorly with more industry-relevant numerical and computational methods. Engineering and mathematics educators agree that there is a need to shape DE curricula to reflect the mathematical practices of the disciplines whose students take thecourse. Current reform efforts such as inquiry-based instruction, modeling-based instruction, and computational tool inclusion are designed to make DE knowledge and methods more transferable to engineering contexts. However, evidence that such efforts result in students successfully harnessing previously learned DE skills in engineering settings is lacking.This study used the preparation for future learning (PFL) and actor-oriented transfer (AOT) frameworks to study the DE problem-framing processes of engineering students. A cohort of 15 chemical and mechanical engineering students and 16 chemical engineering, mechanical engineering, and mathematics faculty members at three universities participated in semi-structured task-based interviews and class observations. Qualitative data analysis was performed using an analytical framework, assembled from themes in the literature review, characterizing the methods, tools, pedagogies, and examples permeating DE and engineering instruction. Faculty and student perspectives were synthesized to characterize the methods of inquiry engineering students experienced intheir mathematics coursework, how mathematical tools are valued differently between the disciplines, and how inqustudents accessed DE knowledge in engineering problem framing. Findings suggest that students' methods of problem framing may relate to their development of a sense of the utility of DE concepts, methods, and tools. Additionally, transfer of DEknowledge into engineering applications may be impacted by such things as the extent to which mathematics and engineering domains provide opportunities for tool selection, the extent to which knowledge domains are artificially compartmentalized, and the reality that many undergraduate engineering students are still developing critical algebra and calculus understandings during and after DE. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com.bibliotheek.ehb.be/en-US/products/dissertations/individuals.shtml.]