Motivating Second-Order Differential Equation Models Beyond the Spring–Mass System

Authors

• Katherine E. Rutherford, United States Military AcademyFollow
• John C. Borger, United States Military AcademyFollow

Publication Date

9-2-2026

Keywords

Spring-mass, Second-order Ordinary Differential Equations, Modeling with Ordinary Differential Equations, Fluid mechanics models

Disciplines

Mathematics | Physical Sciences and Mathematics | Science and Mathematics Education

Abstract

Students in introductory differential equations courses often learn to associate specific equation types with familiar physical systems. While this approach supports procedural fluency, it can inhibit transfer learning: students recognize familiar forms rather than construct models from first principles. This paper presents a modeling-first instructional approach designed to help students recognize the shared structure of second-order differential equations across diverse contexts. Students derive governing equations from fundamental physical laws for multiple systems, including an Inductor, Resistor, and Capacitor (LRC) electrical circuit, a simple pendulum, and viscous pipe flow. Through guided comparison, students identify common components such as inertia, damping, restoring effects, and forcing, and connect these components across models. By emphasizing model construction before classification, this approach encourages students to view differential equations as general representations of relationships rather than isolated formulas. This shift supports transfer learning and helps students approach unfamiliar systems with greater confidence and conceptual understanding.

Recommended Citation

Rutherford, Katherine E. and Borger, John C. (2026) "Motivating Second-Order Differential Equation Models Beyond the Spring–Mass System," CODEE Journal: Vol. 20: Iss. 1, Article 3.
Available at: https://scholarship.claremont.edu/codee/vol20/iss1/3

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