This paper presents MathBode, a dynamic diagnostic tool for mathematical inference of large-scale language models (LLMs). Instead of focusing on one-time accuracy, MathBode treats each parameter as a system, driving a single parameter with a sine wave and fitting the fundamental harmonic response of the model output to the exact solution. This yields interpretable frequency-decomposition metrics, such as gain (amplitude tracking) and phase (delay), which are then formed into a Bode-style fingerprint. MathBode exhibits systematic lowpass behavior and phase delay growth in five closed-form families: linear equations, rate/saturation, compounding, 2x2 linear systems, and pseudo-triangles, which are not detectable by accuracy alone. Various models are compared against a symbolic baseline for instrument calibration ($G \approx 1$, $\phi \approx 0$). The results provide a concise and reproducible protocol that complements standard benchmarks by providing actionable measures of inference fidelity and consistency, distinguishing between leading and intermediate models for dynamics. The dataset and code are made publicly available, enabling further research and adoption.