A Sentence About a Rate

You'll learnto turn a description in English into an equation with a derivative on one side, to attach the constant to the whole phrase that follows "proportional to", and to read a finished model's behaviour without solving it.

A differential equation is a sentence about a rate, written in symbols. Nothing in this topic gets solved, and nothing needs to be: the model is the deliverable. Two moves carry the whole thing. Words like grows, changes and rate of change mean a derivative goes on the left, so a sentence about how fast something moves never becomes an equation about the thing itself. And whatever phrase follows the words proportional to is exactly what the constant multiplies, in full — which is why the parentheses in a cooling law are not decoration but the entire statement.

8 STEPS · 6 QUICK CHECKS · TRANSLATE THE SENTENCE, DO NOT SOLVE IT · v1

rate ⇒ derivative on the left · proportional to ⇒ k times the WHOLE phrase

Before you start

What you're looking at

A ledger. The English phrase sits on the left, an arrow points right, and the equation it produces sits on the right. Blue is the sentence, pink is the equation, violet is a rival reading that does not say the same thing.

The question

Which side of the equation is the rate, and what exactly does the constant multiply? Those two decisions produce almost every model in this unit, and almost every error in this topic.

Watch for

Nothing here is solved. Every scene stops at the model, because the model is what the question asks for.

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