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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn7367Paper 1 · Compulsory contentDifferential equationsFirst-order equations: integrating factors

Differential equations · Subtopic 1 of 2

First-Order Linear Differential Equations

An equation $\frac{dy}{dx}+P(x)y=Q(x)$ can't usually be separated. Multiply through by the integrating factor $I=e^{\int P\,dx}$ : the left side becomes $\frac{d}{dx}(Iy)$ , exactly the derivative of a product, so one integration solves it.

Cambridge 9231 Paper 2Edexcel YFM01 WFM02Edexcel 9FM0 Paper 1Edexcel 9FM0 Paper 2AQA 7367 Paper 1AQA 7367 Paper 2≈ 40 minRevision 1 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Recognise first-order linear equations.
Find the integrating factor $e^{\int P\,dx}$ .
Find general and particular solutions.

Pause and think

What is $\frac{d}{dx}(x^2y)$ , if $y$ is a function of $x$ ?

Section 2 of 14

Core idea

Start hereA tank that fills and leaks

Salt water pours into a tank while the mixture drains out. The amount of salt changes at a rate that depends on how much is already there and on time. You can't separate the variables. But multiply the equation by one clever function and it collapses into a single derivative you can integrate.

The big idea

Write $\frac{dy}{dx}+P(x)y=Q(x)$ . Multiply by $I=e^{\int P\,dx}$ ; the left side becomes $\frac{d}{dx}(Iy)$ . Integrate: $Iy=\int IQ\,dx+c$ .

1. Why the integrating factor works

Making a product rule appear

$I\frac{dy}{dx}+IPy=IQ$
Multiply the equation by some function $I(x)$ .
$\frac{d}{dx}(Iy)=I\frac{dy}{dx}+\frac{dI}{dx}y$
Product rule: we want the left side to equal this.
$\frac{dI}{dx}=IP$
Match the $y$ terms.
$\ln I=\int P\,dx\Rightarrow I=e^{\int P\,dx}$
Separate and integrate (no constant needed).
$\frac{d}{dx}(Iy)=IQ\Rightarrow Iy=\int IQ\,dx+c$
Integrate both sides.

I(x)=e^{\int P(x)\,dx}

$P(x)$: coefficient of $y$ once $\frac{dy}{dx}$ has coefficient $1$ .
$Q(x)$: the right-hand side.

2. Solving

Worked example $\frac{dy}{dx}+\frac yx=3x$ , $y(1)=2$

Find $y$ .

Goal: Particular solution.
Strategy: $I=e^{\ln x}=x$ .

$I=e^{\int\frac1x\,dx}=e^{\ln x}=x$
Integrating factor ( $x>0$ ).
$\frac{d}{dx}(xy)=3x^2$
Multiply through by $x$ .
$xy=x^3+c$
Integrate.
$2=1+c\Rightarrow c=1$
Condition.

Answer

y=x^2+\frac1x

✓ Check

y'=2x-\frac1{x^2}

;

y'+\frac yx=2x-\frac1{x^2}+x+\frac1{x^2}=3x

✓.

Worked exampleDividing first

Solve $\cos x\frac{dy}{dx}+y\sin x=\cos^2x$ , $y(0)=2$ .

Goal: $y$ .
Strategy: Standard form: divide by $\cos x$ .

$\frac{dy}{dx}+y\tan x=\cos x$
Coefficient of $y'$ now $1$ .
$I=e^{\int\tan x\,dx}=e^{\ln\sec x}=\sec x$
$\int\tan x=\ln\sec x$ .
$\frac{d}{dx}(y\sec x)=1\Rightarrow y\sec x=x+c$
$\sec x\cos x=1$ .
$c=2\Rightarrow y=(x+2)\cos x$
Condition.

Answer

y=(x+2)\cos x

✓ Check

y(0)=2

✓.

Common misconceptionConstant placement Add

c

before dividing by

I

y=\frac{\int IQ\,dx+c}{I}

, so

c

gets divided too.

Edge caseSeparable too? If

Q=0

the equation is also separable; both methods agree.

Pause and think

Integrating factor for $\frac{dy}{dx}-\frac{2y}x=x$ ?

Summary

Standard form first.
$I=e^{\int P\,dx}$ .
$\frac{d}{dx}(Iy)=IQ$ ; integrate; add $c$ ; divide.

Integrating factor: Multiplier turning the left side into an exact derivative.
Particular solution: Solution fitting given conditions.

HistoryLeibniz and the Bernoullis. Gottfried Leibniz found the integrating factor method in the 1690s, while the Bernoulli brothers raced to solve related equations.

Fun factCooling coffee with a heater. Newton's law of cooling with a changing room temperature is a first-order linear equation, solved by an integrating factor.

Section 3 of 14

Key formulae

Standard form

\frac{dy}{dx}+P(x)y=Q(x)

Integrating factor

I(x)=e^{\int P(x)\,dx}

Solution

I\,y=\int I\,Q\,dx

TrickDivide first. If the equation starts

x\frac{dy}{dx}+\ldots

, divide by

x

so the coefficient of

\frac{dy}{dx}

1

before finding

P

Section 4 of 14

Worked examples

Example 1 · A particular solution0 / 3 steps

Solve

\frac{dy}{dx}+\frac yx=3x

with

y=2

when

x=1

Example 2 · Exponential factor0 / 3 steps

Solve

\frac{dy}{dx}-2y=4

y(0)=1

Section 5 of 14

Explore the graph

A family of solutions

Loading the interactive graph…

Try this

Every value of $c$ gives a solution of $\frac{dy}{dx}+\frac yx=3x$ .
Choose $c$ so the curve passes through $(1,2)$ : $c=1$ .

General solution $y=x^2+\frac cx$ ; the condition picks one curve.

In the real worldDrug in the bloodstream. A drip adds drug at a constant rate while the body removes it at a rate proportional to the amount:

\frac{dx}{dt}+kx=r

, solved with

e^{kt}

Section 6 of 14

Common mistakes

IncorrectUsing

P

before dividing by the coefficient of

y'

CorrectStandard form first

The method needs

\frac{dy}{dx}

with coefficient

1

IncorrectForgetting

+c

before dividing by

I

Correct

Iy=\int IQ\,dx+c

Adding

c

later gives the wrong dependence on

x

Incorrect

e^{\ln x}

left unsimplified

Correct

=x

Simplify the integrating factor before using it.

dy/dx + P(x) y = Q(x): can't separate

Integrating factor: allow me to turn you into a product rule.

Section 7 of 14

Exam tip

Exam tipShow the product. Write

\frac{d}{dx}(Iy)=IQ

explicitly: it shows the examiner you understand why the method works.

Section 8 of 14

Guided practice

SA-FP-DE1-G012 marksL1 FoundationNon-calculatorIntegrating factor

The integrating factor for

\frac{dy}{dx}+\frac{2y}x=x

x^k

. Find

k

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-FP-DE1-0011 markL1 FoundationNon-calculatorIntegrating factor

The integrating factor for

\frac{dy}{dx}+3y=e^x

e^{kx}

. Find

k

Salam Academy Original Practice · Proposed mark scheme

SA-FP-DE1-0024 marksL3 ExamNon-calculatorSolving

Solve

\frac{dy}{dx}+\frac yx=3x

with

y=2

x=1

. Give

y

in terms of

x

Salam Academy Original Practice · Proposed mark scheme

SA-FP-DE1-0034 marksL3 ExamNon-calculatorSolving

\frac{dy}{dx}-2y=4

y(0)=1

. Find the exact value of

y

when

x=\ln2

Salam Academy Original Practice · Proposed mark scheme

SA-FP-DE1-0042 marksL2 StandardNon-calculatorIntegrating factor

For

x\frac{dy}{dx}+3y=x^2

, which is the integrating factor?

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-FP-DE1-0055 marksL4 ChallengingCalculatorSolving

\cos x\frac{dy}{dx}+y\sin x=\cos^2x

with

y=2

x=0

. Find

y

when

x=\pi

(to 4 s.f.).

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Standard form.
$I=e^{\int P\,dx}$ .
$Iy=\int IQ\,dx+c$ .

Section 12 of 14

Quick quiz

Q1 · SA-FP-DE1-Q011 markL1 FoundationNon-calculatorIntegrating factor

The integrating factor for

\frac{dy}{dx}+P(x)y=Q(x)

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: 7367 A-level Further Mathematics
Where: Paper 1 and 2 · Differential equations
Expected: First-order linear equations using an integrating factor.
Calculator: Permitted on all papers.
Typical style: AQA mixes short multiple-choice questions with longer structured ones.
Open in: 7367 · Paper 1 7367 · Paper 2

Section 14 of 14

Past paper questions

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Salam Academy original content · Content status: reviewed and published. Syllabus mapping: AQA 7367, First teaching 2017 (official specification).