Publication record
- Original publication period
- July–December 2023
- Digitized / uploaded online
- August 10, 2026
The digitization/upload date records when this file was added to the website; it is not the article's original publication date.
Abstract
Students often learn first-order differential equations as a catalogue of solution techniques, making method selection difficult when equations are embedded in engineering contexts. This article develops a decision framework that connects equation structure, solution method, initial conditions, and physical interpretation. Drawing on an instructional worktext in elementary differential equations, the synthesis distinguishes separable, first-order linear, exact, homogeneous-substitution, and Bernoulli forms, then integrates these forms with four application families: exponential growth and decay, Newtonian cooling, well-mixed tanks, and simple RC/RL circuits. The article proposes a sequence of structural tests, demonstrates why several equivalent-looking equations require different transformations, and emphasizes dimensional and limiting-value checks. Its central contribution is a model-method-verification triad that treats differential-equation solving as engineering modeling rather than symbolic manipulation. The framework can guide lesson design, worked examples, and analytic rubrics, while empirical evaluation is reserved for subsequent classroom studies.