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Ideas for teaching differential

equationsSolving differential equations, and using them for

modelling in kinematics and other contexts, is a

key topic in A level Further Mathematics.

Interesting applications such as motion with air

resistance, resonance and predator-prey models

can arouse curiosity and motivate classroom

discussion. In this session we’ll explore some

applications involving simple practical examples

and through the use of technology.

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Ideas for

teaching

differential

equations

Sharon Tripconey

Avril Steele

Ben Sparks

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elastic strap

baby

baby-saddle

Baby Bouncer - modified version

(for use in the classroom)

elastic band

water bottle

bag (with

tie-handles)

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Modelling cycle

Real-world

problem

Simplifying

assumptions

Mathematical

model

(equations etc)

Analysis

and solution

PredictionExperiment

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Setting up a model

Simplifying assumptions:

▪ motion is only up and down

▪ the elastic band is taut and has constant

stiffness

▪ ignore resistive forces

▪ ……

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In equilibrium

acceleration = 0

Not in equilibrium

acceleration ≠ 0

RT

W

W

T0

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Model 1

The forces acting are:

▪ Weight,

▪ Tension, ( ) Nm

k x gk

+

Nmg

displacement

acceleration

0 constant (stiffness)

x

x

k

T

W

x

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Model 1

T

W

x

( )

0

W T mx

mmg k x g mx

k

mx kx

+ − =

− + =

+ =

0k

x xm

+ =

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Simple Harmonic Motion (SHM)

0k

x xm

+ =

> 0 k

m

2 x x= −

2 0x x+ =2 =

k

m

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Simple Harmonic Motion (SHM)2 x x= −

sin( )

sin( ) cos( )

x a t

or

x A t B t

= +

= +

Amplitude: 𝑎 = 𝐴2 + 𝐵2

Period =2𝜋

𝜔

Frequency =𝜔

2𝜋=

1

2𝜋

𝑘

𝑚

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Modelling cycle

Real-world

problem

Simplifying

assumptions

Mathematical

model

(equations etc)

Analysis

and solution

PredictionExperiment

0k

x xm

+ =

sin( )

sin( ) cos( )

x a t

or

x A t B t

= +

= +

𝑘 or 𝑚important?

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Step 1: Suspend from a FIXED point

Step 2: Establish the equilibrium position

(just let it hang to start)

Step 3: Pull the bottle downwards

and release

Step 4: Observe the motion.

kfrequency

m

Step 5: Repeat

Can you change the frequency

of oscillation? If so, how?

ST Version 1.8 27/02/19.

Modelling cycle

Real-world

problem

Simplifying

assumptions

Mathematical

model

(equations etc)

Analysis

and solution

PredictionExperiment

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Revising the model

Simplifying assumptions:

▪ motion is only up and down

▪ the elastic band is taut and has constant

stiffness

▪ ignore resistive forces

▪ ……

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Model 2

The forces acting are:

▪ Weight,

▪ Tension,

▪ Resistance to motion,

( ) Nm

k x gk

+

Nmg T

W

xR

displacement

velocity

acceleration

0 constant (stiffness)

> 0 constant (drag)

x

x

x

k

Nx

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Model 2

T

W

xR

( )

0

W R T mx

mmg x k x g mx

k

mx x kx

− − =

− − + =

+ + =

0k

x x xm m

+ + =

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Using technology to explore

www.geogebra.org/m/ej8cmqjx

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Tacoma Narrows Bridge

Collapse

See image

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Referring to requirements for A level FM

I 7 Solve the equation for simple harmonic motion

ሷ𝑥 = −𝜔2𝑥 and relate the solution to the motion

I 8 Model damped oscillations using 2nd order

differential equations and interpret their solutions

I 9 Analyse and interpret models of situations with one

independent variable and two dependent variables

as a pair of coupled 1st order simultaneous

equations and be able to solve them, for example

predator-prey models

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Predator – prey

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Predator – prey ▪ Using technology to explore

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In general, a pair of first order linked differential equations looks like

𝑑𝑥

𝑑𝑡= 𝑓 𝑥, 𝑦, 𝑡

𝑑𝑦

𝑑𝑡= 𝑔(𝑥, 𝑦, 𝑡)

For a predator-prey example, a typical pair is

𝑑𝑥

𝑑𝑡= 𝛼𝑥 − 𝛽𝑥𝑦

𝑑𝑦

𝑑𝑡= 𝛿𝑥𝑦 − 𝛾𝑦

where 𝛼, 𝛽, 𝛾, 𝛿 are positive constants.

This is the LOTKA-VOLTERRA model.

Coupled DEs

This non-

linear pair

can be

solved

analytically

but it’s not

within the

scope of A

level FM

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Systems of linear first order DEs

In A level FM, the pair of simultaneous equations

involving rates of change will be of the form:

(𝑎, 𝑏, 𝑐 and 𝑑 are constants)

( )

( )

dxax by f t

dt

dycx dy g t

dt

= + +

= + +

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The simultaneous differential equations

model the quantities of two compounds, 𝑥 and 𝑦,

produced in a chemical reaction.

How do the quantities vary over time?

d d4 6 28 3 2 26

d d= − + + = − + +

x yx y x y

t t

Chemical reaction

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Chemical reaction▪ Using technology to explore

www.geogebra.org/m/pwwcdafy

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Applications

First order differential equations can be used

in modelling:

▪ Exponential growth

▪ Radioactive decay

▪ Cooling liquids or objects

▪ Continuous compound interest (stocks,

investments)

▪ Mixing solutions

▪ …..

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Applications

Second order differential equations can be used

to model:

▪ Falling objects

▪ Mechanical oscillations

• Damped oscillations

• Forced oscillations

• Resonance

▪ Electrical circuits

• Series LCR circuits for radio and

communication engineering

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A pair of coupled first

order differential equations

can be used to model:

▪ Spread of epidemics; as

people transform from

being “at risk” to “infected”

to immune”

▪ Predator-prey (e.g. foxes

and rabbits, bears and

berries)

▪ Competing populations:

• Two newspapers of similar

type compete for the same

potential circulation

• Two bird-of-prey species

hunt a woodland for the

same small mammals

Applications

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The SIR model is used to model infectious disease

(e.g. Zombies)

Susceptible

Infected

Recovered

3 variables, so need 3 Differential Equations

Zombie horde...?

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Herd Immunity and Vaccination

The SIR model can be used to discuss effectiveness of

vaccines and how even those who cannot be vaccinated

might be protected via Herd Immunity.

All modelled with an SIR model and differential equations.

Excellent introduction here: http://op12no2.me/toys/herd/

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About the AMSP▪ A government-funded initiative, managed by MEI,

providing national support for teachers and students in

all state-funded schools and colleges in England.

▪ It aims to increase participation in AS/A level

Mathematics and Further Mathematics, and Core

Maths, and improve the teaching of these qualifications.

▪ Additional support is given to those in priority areas to

boost social mobility so that, whatever their gender,

background or location, students can choose their best

maths pathway post-16, and have access to high quality

maths teaching.

amsp.org.uk

2019 FM Conference

V1.2

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Differential equations

A level Mathematics content

G6 Construct simple differential equations in pure mathematics and in context (contexts may include kinematics, population growth and modelling the relationship between price and demand)

H7 Evaluate the analytical solution of simple first order differential equations with separation of variables, including finding particular solutions (Separation of variables may require factorisation involving a common factor)

(Reference: DFE-00706-2014 Mathematics AS and A level content)

A level Further Mathematics content

I1 Find and use an integrating factor to solve differential equations of form

𝑑𝑦

𝑑𝑥+ 𝑃(𝑥)𝑦 = 𝑄(𝑥) and recognise when it is appropriate to do so

I2 Find both general and particular solutions to differential equations

I3 Use differential equations in modelling in kinematics and in other contexts

I4 Solve differential equations of form 𝑦′′ + 𝑎𝑦′ + 𝑏𝑦 = 0 where 𝑎 and 𝑏 are constants by using the auxiliary equation

I5 Solve differential equations of form 𝑦′′ + 𝑎𝑦′ + 𝑏𝑦 = 𝑓(𝑥) where 𝑎 and 𝑏 are constants by solving the homogeneous case and adding a particular integral to the complementary function (in cases where f(𝑥) is a polynomial, exponential or trigonometric function)

I6 Understand and use the relationship between the cases when the discriminant of the auxiliary equation is positive, zero and negative and the form of solution of the differential equation

I7 Solve the equation for simple harmonic motion �̈� = −𝜔2𝑥 and relate the solution to the motion

I8 Model damped oscillations using 2nd order differential equations and interpret their solutions

I9 Analyse and interpret models of situations with one independent variable and two dependent variables as a pair of coupled 1st order simultaneous equations and be able to solve them, for example predator-prey models

(Reference: DFE-00707-2014 Further mathematics AS and A level content)

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What is a differential equation?

If you are using maths to describe the spread of diseases, the growth of a flower or fluctuations in the financial markets you will soon find that you’ll need to be using differential equations. Differential equations are fundamental to nearly all modern applications of mathematics and many natural phenomena. The vast majority of differential equations cannot be solved using known analytical methods; solutions are found using numerical methods. Essentially, a differential equation is an equation involving one or more derivatives. Here are some examples:

Terminology

(A) 1st order, linear – we can solve it (integrate both

sides)

(B) 1st order, linear – we can solve it (separate variables and integrate)

(C) 1st order, linear – we can solve it (use

integrating factor)

(D) 2nd order, linear, constant coefficients – we can solve it (auxiliary equation method)

(E) 1st order, non-linear – we can’t solve it

analytically (use a numerical method e.g. Euler’s method)

(F) 2nd order, linear, variable coefficients – we

(probably) can’t solve it analytically (use a numerical method)

(G) 2nd order, non-linear, constant coefficients – we

can’t solve it analytically (use a numerical method)

When f(x) = 0, the

equation is

homogeneous

When f(x) ≠ 0, the

equation is

non-homogeneous

2

2 2

2

2

2

22

2

22

2

(A) 2

(B) sin

(C) 3

(D) 6 3 cos

(E)

(F) 3 2 sin

(G)

x

x

x

dyx x

dx

dyy x

dx

dyx y e

dx

d y dyy x

dxdx

dyy x e

dx

d y dyx x y e x

dxdx

d y dyy x

dxdx

2

25 6 4 xd y dy

y edx dx

The r.h.s.

is of form

f(x)

It is

“second

order”

It is

“linear”

It has

“constant

coefficients”

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Notation Sometimes, especially when we have a variable that changes with time, we use the more compact ‘dot’

notation in which �̇� is d𝑥

d𝑡 and �̈� is

d2𝑥

d𝑡2.

Damped oscillations

𝑎�̈� + 𝑏�̇� + 𝑐𝑥 = 0 𝑎 > 0

Roots of the auxiliary equation are:

𝜆1 =−𝑏+√𝑏2−4𝑎𝑐

2𝑎 and 𝜆2 =

−𝑏−√𝑏2−4𝑎𝑐

2𝑎

When both roots are real and negative, or the real part of complex roots is negative, the motion is

damped and the curve eventually tends to zero. In these cases 𝑏 > 0

When 𝑏2 − 4𝑎𝑐 < 0 we have under-damping (solution contains sin / cos functions)

When 𝑏2 − 4𝑎𝑐 = 0 we have critical-damping (solution doesn’t contain sin / cos)

When 𝑏2 − 4𝑎𝑐 > 0 we have over-damping (solution doesn’t contain sin / cos)

When 𝑏 = 0 the motion is

Simple Harmonic Motion (S.H.M)

(constant amplitude)

When 𝑏 < 0 and 𝑏2 − 4𝑎𝑐 < 0 the oscillations increase in

amplitude, being enveloped in a positive exponential curve.

𝑥 = 𝐴𝑒𝜆1𝑡 + 𝐵𝑒𝜆2𝑡

under-damping

critical-damping

under-damping

over-damping

Note it is possible to have b>0 and

no damping

e.g. �̈� + �̇� − 2𝑥 = 0

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Coupled differential equations

Problem:

In the simultaneous differential equations

d d

4 6 28 3 2 26d d

x y

x y x yt t

𝑥 and 𝑦 are the quantities of compounds produced in a chemical reaction.

(i) Eliminate y from the equations to show that 2

2

d d2 10 100

d d

x xx

t t.

(ii) Find the general solution for 𝑥 and use this to find the corresponding general solution for 𝑦.

(iii) Given that 𝑥 = 𝑦 = 0 when 𝑡 = 0, find particular solutions for 𝑥 and 𝑦.

(iv) Sketch the solution curves, indicating the long term values for 𝑥 and 𝑦. Explain how the long

term values could be found without solving the differential equations.

A worked solution:

2

2

d4 6 28

d

1 d4 28

6 d

d 1 d d4

d 6 d d

xx y

tx

y xt

y x x

t t t

Substituting into d

3 2 26d

yx y

t:

2

2

2

2

2

2

1 d d 1 d4 3 2 4 28 26

6 d d 6 d

d d d4 18 2 8 56 156

d d d

d d2 10 100

d d

x x xx x

t t t

x x xx x

t t t

x xx

t t

(ii) Auxiliary equation is

2 2 10 0

2 4 40 2 61 3

2 2

ii

Complementary function is e ( cos3 sin 3 )tx A t B t

Particular integral is x c

Substituting into differential equation gives 10 100 10c c

General solution for x is e ( cos3 sin 3 ) 10tx A t B t

d

e ( 3 sin 3 3 cos3 ) e ( cos3 sin 3 )d

t txA t B t A t B t

t

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16

12

1 d4 28

6 d1e ( 3 sin 3 3 cos3 ) e ( cos3 sin 3 )

6

4e ( cos3 sin 3 ) 40 28

e ( 3 4 )sin 3 (3 4 )cos3 2

e ( )sin 3 ( )cos3 2

t t

t

t

t

xy x

t

A t B t A t B t

A t B t

A B B t B A A t

A B t B A t

(iii)When t = 0, x = 0 0 10 10A A

When t = 0, y = 0 120 ( ) 2 4 6B A A B B

e ( 10cos3 6sin 3 ) 10tx t t

12 e 16sin3 4cos3 2

e 8 sin3 2cos3 2

t

t

y t t

t t

(iv)

The long term values could be found by solving the equations d

0d

x

t and

d0

d

y

t

simultaneously.

2 4 6

5

10

15

20

t

x

1 2 3 4 5 6

5

10

t

y

2