https://www.chenk.top/en/tags/mathematical-modeling/Recent content in Mathematical Modeling on Chen Kai BlogHugoenSat, 01 Jul 2023 09:00:00 +0000https://www.chenk.top/en/ode/01-origins-and-intuition/Sat, 01 Jul 2023 09:00:00 +0000https://www.chenk.top/en/ode/01-origins-and-intuition/<p><strong>Everything around you is changing.</strong> Coffee cools, populations grow, pendulums swing, viruses spread, stocks oscillate, planets orbit. None of these systems are described by <em>what something equals</em> — they are described by <em>how fast something changes</em>. That second mode of description is what differential equations are for, and learning to read them is, quite literally, learning to read the language physics and biology are written in.</p> <p>This chapter rebuilds your intuition from scratch. We start with a single cup of coffee, derive the same equation that governs radioactive decay and capacitor discharge, then climb upward to direction fields, classification, and the existence-and-uniqueness theorem that tells you when an ODE has a sensible answer at all.</p>