Learn9MA0Paper 2 · Pure Mathematics 2IntegrationForming differential equations

Integration · Subtopic 11 of 14

Forming Differential Equations

Turn words about change into mathematics. "Grows in proportion to", "decreases at a rate inversely proportional to", "flows in and leaks out": each becomes an equation for a derivative, ready to solve.

Cambridge 9709 Paper 3Edexcel YMA01 WMA14Edexcel 9MA0 Paper 1Edexcel 9MA0 Paper 2AQA 7357 Paper 1AQA 7357 Paper 2≈ 30 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Translate a description of a rate of change into a differential equation.
Use a negative sign for a decreasing quantity and introduce a constant of proportionality.
Find the constant from a rate given at one moment.
Combine rates using the chain rule, and inflow minus outflow.

Pause and think

A cup of tea at $90^\circ$ C is left in a $20^\circ$ C room. Does it cool faster at the start or after half an hour? Why?

Section 2 of 14

Core idea

Start hereLaws about change

Nature rarely tells you a formula directly. It tells you how things change: "a population grows in proportion to its size", "hot tea cools faster when it's much hotter than the room". Written in symbols, each is a differential equation, an equation containing a derivative.

The big idea

Translate words into rates: "rate of change of $N$ " is $\frac{dN}{dt}$ ; "proportional to" is " $=k\times$ "; "decreasing" means a negative rate. Net rate = rate in − rate out. Use the chain rule to link different variables.

1. The language

Growth where the gradient equals half the height

t = 0.30

\dfrac{dN}{dt}=\tfrac12N

. As

N

gets bigger, the tangent gets steeper: the more there is, the faster it grows.

Worked exampleTranslating

Write differential equations for three statements.

Goal: Equations.
Strategy: Name the variable; translate phrase by phrase.

$\frac{dP}{dt}=kP$
Population grows at a rate proportional to its size.
$\frac{d\theta}{dt}=-k(\theta-20)$
Cools at a rate proportional to the excess over 20 °C; minus because decreasing.
$\frac{dy}{dx}=kxy^2$
Gradient proportional to $xy^2$ .

AnswerAs shown (

k>0

✓ CheckEach has a derivative on the left and the stated dependence on the right ✓.

2. Rate in minus rate out

Worked exampleLeaky tank

Water enters at 5 litres/min and leaks at $0.1V$ litres/min. Form the equation and find the steady volume.

Goal: Equation; equilibrium.
Strategy: Net rate.

$\frac{dV}{dt}=5-0.1V$
In minus out.
$\frac{dV}{dt}=0\Rightarrow V=50$
Steady state: no change.

Answer

\frac{dV}{dt}=5-0.1V

;

V\to50

litres.

✓ CheckAt

V=50

, leak

=5=

inflow ✓.

3. Chain rule links

Worked exampleMelting snowball

A spherical snowball's volume decreases at a rate proportional to its surface area. Show its radius decreases at a constant rate.

Goal: $\frac{dr}{dt}$ constant.
Strategy: Write the statement, then use $\frac{dV}{dt}=\frac{dV}{dr}\frac{dr}{dt}$ .

$\frac{dV}{dt}=-k(4\pi r^2)$
Statement.
$V=\tfrac43\pi r^3\Rightarrow\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$
Chain rule.
$4\pi r^2\frac{dr}{dt}=-4\pi kr^2\Rightarrow\frac{dr}{dt}=-k$
Equate; cancel.

AnswerConstant rate

-k

∎.

✓ CheckUnits: length/time ✓.

Pause and think

Write: the rate of decrease of $m$ is proportional to $\sqrt m$ .

Summary

Words → rates.
In − out.
Chain rule between variables.

Differential equation: An equation involving a derivative.

HistoryNewton and a hot iron. In 1701 Isaac Newton published, anonymously, a short paper on heating a block of iron in a fire and timing how it cooled. He proposed that the rate of cooling is proportional to the difference in temperature from the surroundings. "Newton's law of cooling" is still the first model taught in physics and maths.

Keep reading for free

Create a free account to read the rest of this lesson: key formulae, worked examples, explore the graph, common mistakes, exam tip, guided practice and more. Your answers are checked and your progress is saved.

Create free account Sign in