Ryan Collison

Linear first-order differential equations look messy until you multiply both sides by the right function. That integrating factor makes the left side a perfect derivative — suddenly the equation almost solves itself.

Separable equations are the first class of differential equations most students can actually solve by hand — and the technique reveals something important about the structure of ODEs. Recognizing when variables can be separated is a skill that transfers to much harder problems.

Growth, decay, cooling, mixing — one simple differential equation models them all. First-order ODEs are where calculus meets the real world, and they're more powerful than they first appear.

The split between ordinary differential equations (ODEs) and partial differential equations (PDEs) isn't just mathematical bookkeeping. It's the difference between modeling change along one dimension versus modeling change across space and time simultaneously. ODEs describe trajectories. PDEs describe fields. That distinction unlocks entirely different classes of

Newton's second law, the heat equation, the Schrödinger equation — all of them are differential equations. Instead of describing a quantity directly, they describe how it changes. That shift in perspective is one of the most productive moves in all of mathematics.

Most of physics, biology, and economics is written in differential equations. The insight is simple but powerful: instead of asking where something is, ask how fast it's moving and in what direction — then integrate forward.