The Language of Change

Differential Equations · Axiom Academy

How a sentence about change becomes a differential equation — and what it means to solve one. 1. A Rate of Change Is a Slope Whenever we say a quantity is "changing," we mean its graph against time has a slope at that instant. Watch the tangent line ride along the curve below — its tilt at any moment IS the rate of change right then. The rate of change of a quantity Q with respect to time Here's the translation in action. A cooling cup of coffee obeys a simple verbal rule: "the temperature drops at a rate proportional to how much hotter it is than the room." Watch that sentence become a differential equation. 3. What Does "Solving" Actually Mean? Solving an algebra equation like 2x+3=11 gives you a number . Solving a differential equation gives you a function — and usually a whole family of them, one for every possible starting condition. Watch the family of cooling curves unfold below, each one satisfying , just starting from a different initial temperature. The general solution — one equation, an entire family of curves Checking a solution by substitution To verify really solves the equation, differentiate it: . Now substitute both into the right-hand side: . The two sides match exactly — this is how you confirm ANY claimed solution to a differential equation: differentiate it and plug back in. 4. Classifying: Order and Linearity

This is the written version of the interactive lesson above. See the full Differential Equations course.