Almost every differential equation in science arrives the same way: somebody says how fast something changes, in terms of how much of it there is. "Twice as much, twice as fast" is a proportionality and becomes one line of mathematics. Getting the model right is mostly getting the signs right, and the way to check a sign is to ask what the equation predicts when the quantity is large.

Translating a sentence into an equation is like translating between languages with different word orders: the individual words are easy and the mistake is always in the arrangement. "Cools at a rate proportional to the difference from the room" has to come out with a minus sign, or the coffee heats up.

Newton's law of cooling, $T'=-k(T-A)$, from above and from below. Both approach the room temperature $A$ and neither reaches it; the sign of $k$ is what makes them approach rather than run away.

A model names a quantity, says what its rate of change is, and states the value at one moment. "The rate of change of $P$ is proportional to $P$" is $P'=kP$; "proportional to the difference between $T$ and the surroundings , and always towards " is with . The constant is named, not chosen, and is fixed later by a measurement.

• Proportional to means multiplied by a constant: $y'=ky$. Inversely proportional to means divided by it: $y'=k/y$.

A rate proportional to the amount is the amount multiplied by a constant. Which constant is a question for a measurement, not for the modelling.

Put a large value into the right-hand side and ask whether the prediction is what the world does. A hot object must cool.

Great work! You now know how to:

• turn a sentence about a rate into a differential equation
• check the sign of a model by asking what it predicts for a large value
• write a balance law as rate in minus rate out
• use units to catch a mistake before solving anything