Chapter 1: First-Order Equations

A differential equation is a sentence about change: it tells you how fast a quantity is moving at every moment, and asks you to reconstruct where the quantity goes. The simplest such sentences involve only the quantity and its first derivative — a population and its growth rate, a bank balance and its interest, a hot object and how fast it cools. This chapter is about reading those sentences, solving the ones that can be solved by hand, and — just as important — extracting the story an equation tells even before it is solved.

Two ideas organize everything that follows. The first is that a differential equation never has just one solution: it has a family of them, one for each possible starting value, and an initial condition is what picks a single member out of the family. The second is the idea of an equilibrium — a constant solution where the rate of change is zero — and the discovery that equilibria come in stable and unstable kinds, attract or repel their neighbors, and can collide and annihilate when a knob in the model is turned too far.

Every figure in this chapter is interactive. The sliders are the constants of the model — an initial value, a growth rate, a harvest quota — and the whole point is to drive them yourself: predict what a slider will do before you move it, then move it and check. By the end of the chapter you will have watched a solution family sweep through its possibilities, seen a population saturate at carrying capacity, and pushed a harvested ecosystem over the edge of collapse.