2026 ASEE Annual Conference & Exposition

We propose and outline here a restructuring of the traditional undergraduate first course in Differential Equations. With engineering students as its target audience, we place early and targeted emphasis on linear algebra concepts in context and in service of a systems-first approach. Unlike traditional treatments, here all higher-order ordinary differential equations are systematically reduced to first-order systems, enabling students to approach problems uniformly, regardless of order. Linear algebra—including eigenvalues, eigenvectors, fundamental sets, diagonalization, and matrix exponentiation—is introduced and used repeatedly as an essential analytical tool for solving constant coefficient linear systems, which typically account for at least two-thirds of the course content. Unifying our treatment of linear constant coefficient systems is the fundamental matrix e^(At). In addition to not requiring that the coefficient matrix be diagonalizable, e^(At) links integrating factor, variation of parameters, and convolution in ways that are meaningful to engineering applications. Nonlinearity is shown to be sometimes treatable with Calculus, but in general requires linearization techniques to return to the ability to apply linear algebraic tools. This integrated approach enhances students’ ability to analyze coupled systems, assess stability, and apply techniques to mechanical vibrations, electrical circuits, and emerging applications in data-driven engineering. Early assessment indicates improved conceptual understanding, stronger connections between mathematical theory and engineering practice, and greater confidence in applying linear algebra in dynamic system analysis. This work contributes new practices to the field of engineering education by demonstrating that a unified, systems-oriented approach provides a more coherent and application-ready foundation for engineering undergraduates.

The central claim of this work is that the matrix exponential e^(At) provides a unifying analytical framework that can replace the traditional “method-per-topic” structure of introductory differential equations.