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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 2 · Compulsory contentDifferential equationsSecond-order equations: auxiliary equation

Differential equations · Subtopic 2 of 2

Second-Order Linear Differential Equations

For $ay''+by'+cy=f(x)$ , try $y=e^{mx}$ in the "homogeneous" equation: $am^2+bm+c=0$ (the auxiliary equation). Its roots give the complementary function. Add a particular integral matching $f(x)$ , then fix the two constants with two conditions.

Cambridge 9231 Paper 2Edexcel YFM01 WFM02Edexcel 9FM0 Paper 1Edexcel 9FM0 Paper 2AQA 7367 Paper 1AQA 7367 Paper 2≈ 50 minRevision 1 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Form and solve the auxiliary equation.
Write complementary functions for distinct real, repeated and complex roots.
Find particular integrals for polynomial, exponential and trigonometric right-hand sides.
Use two conditions to find a particular solution.

Pause and think

Which function is its own derivative up to a constant multiple?

Section 2 of 14

Core idea

Start hereBouncing on a spring

A mass on a spring with a damper obeys $m\ddot x+c\dot x+kx=0$ : acceleration depends on position and velocity. Will it oscillate, or creep back to rest? The answer depends on the roots of an ordinary quadratic.

The big idea

$ay''+by'+cy=f(x)$ . Auxiliary equation $am^2+bm+c=0$ gives the complementary function; add a particular integral for $f(x)$ ; use two conditions for the constants.

1. The auxiliary equation

Trying

y=e^{mx}

$y=e^{mx},\ y'=me^{mx},\ y''=m^2e^{mx}$
Exponentials keep their shape.
$(am^2+bm+c)e^{mx}=0$
Substitute into $ay''+by'+cy=0$ .
$am^2+bm+c=0$
$e^{mx}\neq0$ .

Each root gives a solution; linear combinations are also solutions.

Complex roots become trig

$m=p\pm qi$
Negative discriminant.
$y=e^{px}(Ce^{iqx}+De^{-iqx})$
Two complex exponential solutions.
$=e^{px}(A\cos qx+B\sin qx)$
Euler: $e^{i\theta}=\cos\theta+i\sin\theta$ ; regroup constants.

\text{CF}=\begin{cases}Ae^{\alpha x}+Be^{\beta x}\\(A+Bx)e^{\alpha x}\\e^{px}(A\cos qx+B\sin qx)\end{cases}

$\alpha,\beta$: distinct real roots.
$\alpha$ (repeated): the extra $x$ gives a second independent solution.
$p\pm qi$: complex roots.

2. Particular integrals

Worked example $y''-3y'+2y=4$

General solution.

Goal: $y$ .
Strategy: CF from $m^2-3m+2=0$ ; PI a constant.

$m=1,2\Rightarrow\text{CF}=Ae^x+Be^{2x}$
Factorise $(m-1)(m-2)$ .
$y=\lambda\Rightarrow2\lambda=4\Rightarrow\lambda=2$
Derivatives of a constant vanish.

Answer

y=Ae^x+Be^{2x}+2

✓ Check

y=2

0-0+4=4

✓.

Worked example $y''-y=0$ , $y(0)=2$ , $y'(0)=0$

Particular solution.

Goal: $y$ .
Strategy: CF; two equations for $A$ , $B$ .

$y=Ae^x+Be^{-x}$
$m=\pm1$ .
$A+B=2,\ A-B=0$
$y(0)$ and $y'(0)$ .
$A=B=1$
Solve.

Answer

y=e^x+e^{-x}=2\cosh x

✓ Check

y''=y

✓.

Common misconceptionPI already in the CF If

f(x)=e^{x}

and

1

is a root, try

\lambda xe^x

(or

\lambda x^2e^x

for a repeated root).

Key takeawayDamping For

\ddot x+2k\dot x+\omega^2x=0

k<\omega

light (oscillates),

k=\omega

critical,

k>\omega

heavy.

Pause and think

What does $y''+9y=0$ describe?

Summary

Auxiliary equation from $e^{mx}$ .
Three CF cases.
PI from the form of $f(x)$ .
GS = CF + PI; two conditions.

Complementary function: General solution with $f(x)=0$ .
Particular integral: One solution with the given $f(x)$ .

HistoryEuler. Leonhard Euler solved linear equations with constant coefficients in 1743, introducing the trial solution

e^{mx}

Fun factCar suspension. A car's springs and dampers obey

my''+cy'+ky=0

. Engineers tune

c

so the auxiliary equation has a repeated root: critical damping, the smoothest ride.

Section 3 of 14

Key formulae

Auxiliary equation

am^2+bm+c=0

Complementary function

\begin{cases}Ae^{\alpha x}+Be^{\beta x}&\text{distinct real}\\(A+Bx)e^{\alpha x}&\text{repeated}\\e^{px}(A\cos qx+B\sin qx)&m=p\pm qi\end{cases}

General solution

y=\text{CF}+\text{PI}

TrickTrial PIs.

f(x)=k

: try

\lambda

. Linear:

\lambda x+\mu

ke^{nx}

\lambda e^{nx}

(or

\lambda xe^{nx}

n

is a root).

\sin

\cos

\lambda\cos+\mu\sin

Section 4 of 14

Worked examples

Example 1 · Distinct roots0 / 3 steps

Solve

y''-3y'+2y=4

Example 2 · Complex roots0 / 2 steps

Solve

y''+2y'+5y=0

Section 5 of 14

Explore the graph

Damped oscillation

Loading the interactive graph…

Try this

Roots $-p\pm qi$ give $e^{-px}\cos qx$ : oscillation inside a decaying envelope.
Set $p=0$ : undamped SHM.

Complex roots: the real part sets decay, the imaginary part sets frequency.

In the real worldElectrical circuits. An inductor, resistor and capacitor in series satisfy

Lq''+Rq'+\frac qC=V(t)

: the same equation as a damped spring.

Section 6 of 14

Common mistakes

IncorrectPI

\lambda e^{x}

when

1

is a root

CorrectTry

\lambda xe^x

\lambda e^x

is already in the CF, so it gives zero.

IncorrectApplying conditions to the CF only

CorrectUse the full general solution

The PI contributes to

y

and

y'

Incorrect

e^{\pm qi}

left in the answer

CorrectWrite

A\cos qx+B\sin qx

Real solutions are expected.

Second-order ODE: looks terrifying

Auxiliary equation: it's just a quadratic, mate.

Section 7 of 14

Exam tip

Exam tipTwo constants, two conditions. Differentiate the general solution before substituting

y'(0)

; find the PI first so it's included.

Section 8 of 14

Guided practice

SA-FP-DE2-G012 marksL1 FoundationNon-calculatorComplementary function

Find the roots of the auxiliary equation for

y''-5y'+6y=0

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-FP-DE2-0012 marksL2 StandardNon-calculatorComplementary function

The general solution of

y''+4y=0

A\cos\omega x+B\sin\omega x

. Find

\omega

Salam Academy Original Practice · Proposed mark scheme

SA-FP-DE2-0021 markL2 StandardNon-calculatorComplementary function

Find the repeated root of the auxiliary equation for

y''+2y'+y=0

Salam Academy Original Practice · Proposed mark scheme

SA-FP-DE2-0032 marksL2 StandardNon-calculatorParticular integral

Find the constant particular integral of

y''-3y'+2y=4

Salam Academy Original Practice · Proposed mark scheme

SA-FP-DE2-0043 marksL3 ExamNon-calculatorParticular integral

Find a particular integral of

y''+y=3x

, in terms of

x

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-DE2-0055 marksL4 ChallengingNon-calculatorParticular solutions

y''-y=0

y(0)=2

y'(0)=0

. Find the exact value of

y

when

x=\ln2

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Auxiliary equation $am^2+bm+c=0$ .
Three CF forms.
PI matches $f(x)$ .
GS = CF + PI; two conditions.

Section 12 of 14

Quick quiz

Q1 · SA-FP-DE2-Q011 markL1 FoundationNon-calculatorComplementary function

If the auxiliary equation has roots

p\pm qi

, the complementary function is

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: Second-order equations with constant coefficients; SHM and damping.
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

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