Knowledge, Innovation and Technology in Scientific Research Vol. 1

Teaching the Galerkin method in engineering courses can be challenging because students must coordinate symbolic derivation, boundary conditions, approximation spaces, and interpretation of numerical error. This article presents a design-and-demonstration proposal integrating GeoGebra and Claude as complementary resources for teaching a Galerkin approximation in an engineering differential equations course. The paper does not report a controlled classroom intervention or measured learning outcomes; instead, it develops and verifies an illustrative instructional workflow. For the boundary-value problem y'' + y = -ax, y(0) = y(1) = 0, the exact solution and a two-term Galerkin approximation are derived, visualised dynamically in GeoGebra, and explained through a Claude-supported interaction. With a = 0.5, the approximation yields a maximum absolute error of approximately 1.52 × 10^-4 near x = 0.516, consistent with the graphical output. The design draws on multimedia-learning principles and literature on dynamic mathematics environments and generative artificial intelligence. GeoGebra supports coordinated symbolic-graphical inspection and parameter exploration, whereas Claude can provide stepwise verbal explanations and prompts for reflection. The contribution lies in coordinating these functions around an established Galerkin example rather than proposing a new numerical method. Because generative AI can produce imprecise terminology or mathematically unreliable output, teacher oversight and verification remain essential. Empirical studies are needed to determine effects on learning, reasoning, and student autonomy.