Concept preview

What you will be able to do

1. 01Translate a described situation into a differential equation, recognising that "the rate of change is proportional to the amount" becomes $\dfrac{dy}{dt}=ky$.
2. 02Set up Newton's law of cooling as $\dfrac{dT}{dt}=k(T-A)$ and mixing or tank problems as rate in minus rate out, then solve for the proportionality constant from given data.
3. 03Interpret what a differential equation says physically and attach correct units to every rate, distinguishing a constant (linear) rate from a proportional (exponential) rate.

A differential equation relates a function to its own derivative. The derivative $\dfrac{dy}{dt}$ is a rate of change, so a differential equation is just an equation that contains a rate. The whole skill of modeling is reading a situation, identifying what is changing and how its rate depends on the current state, and then writing that dependence as an equation. You are not asked to solve it yet — only to state it correctly.