7.1 Modeling with Differential Equations

What they are

A differential equation relates a function to its derivative(s). They model situations where a rate depends on the quantity itself — for example, growth proportional to the current amount.

Translating words

'The rate of change of P is proportional to P' becomes dP/dt = kP, where k is the proportionality constant. This models exponential growth (k > 0) or decay (k < 0).

Key terms

Differential equation

An equation relating a function to its derivative(s).

Proportional rate

'Rate proportional to amount' → dy/dt = ky.

dy/dt = ky

Models exponential growth/decay.

k

The proportionality (growth) constant.

Modeling

Translate a described rate into an equation.

Try an AP-style question

A population grows at a rate proportional to its size. Which equation models this?

1. dP/dt = kP
2. dP/dt = k
3. dP/dt = kt
4. P = kt