Owl: A differential equation can encode a story: how a population grows, how a tank mixes, how coffee cools. The DE is the mechanism; solutions predict short-run paths and long-run settling points.
Picture a lake with fish that breed quickly at first, then level off. Pure exponential growth forgets the limited food — the missing piece is a carrying capacity term. Logistic models remember it.
We'll write DEs from stories, solve or read them qualitatively, and interpret equilibria. Two traps: assuming exponential forever, and ignoring carrying capacity when the story clearly has a ceiling.