Forming Differential Equations (Pure Mathematics 3 – 9709) — Mathematics

Introduction

Many real-world quantities change over time or in response to other variables: a population grows, a liquid cools, a radioactive substance decays. Describing these situations mathematically requires converting a written or verbal statement into a differential equation — an equation that involves a derivative such as $\dfrac{dy}{dx}$ or $\dfrac{dN}{dt}$ .

In 9709 Paper 3, forming differential equations is the essential first step before any solving (separation of variables) takes place. Marks are specifically awarded for correctly setting up the equation, so precision here is vital. This subtopic covers exactly how to read a rate-of-change statement and write it as a differential equation, including when and how to introduce a constant of proportionality $k$ .

Core Concept

Reading a Rate-of-Change Statement

A rate of change is always a derivative. The key phrases to recognise are:

Verbal phrase	Mathematical meaning
"the rate of change of $y$ with respect to $x$ "	$\dfrac{dy}{dx}$
"the rate of increase of $N$ with respect to $t$ "	$\dfrac{dN}{dt}$
"the rate at which $\theta$ decreases"	$-\dfrac{d\theta}{dt}$ (negative, since decreasing)
" $y$ increases at a rate proportional to $y$ "	$\dfrac{dy}{dt} = ky$
"the rate of change is proportional to the square of $x$ "	$\dfrac{dy}{dt} = kx^2$

Direction: Increase or Decrease?

If the quantity is increasing, the derivative is positive.
If the quantity is decreasing, the derivative is negative. This is handled by writing a negative sign explicitly, so that $k$ itself can be taken as a positive constant.

The Constant of Proportionality

When a rate is proportional to some expression $f$ , you write:

\frac{dy}{dt} = k\,f

The constant $k$ is unknown until initial or boundary conditions are used. In exam questions, you may be asked to:

Form the differential equation (introduce $k$ ).
Solve it (separate and integrate).
Evaluate $k$ using a given condition.

All three steps may appear in a single question, but this note focuses on step 1.

Units and Variable Choice

Always define your variables clearly. If the problem says "the population $P$ at time $t$ years", your derivative is $\dfrac{dP}{dt}$ . Matching variable names to the problem avoids sign errors and lost method marks.

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