Using technology

to investigate

differential

equations

(FPT and more)

Tom Button

tom.button@mei.org.uk

mei.org.uk/fpt

FPT

Further Pure with Technology is an A level Further

Maths Option with three topics:

• Investigation of curves (graphing/CAS)

• Number Theory (programming language)

• Differential Equations (graphing/CAS/spreadsheet)

More details:

mei.org.uk/fpt

Sample assessment

Differential Equations in FPT

Differential equations is a new topic in FPT.

Students investigate DEs:

– Using slope fields in graphing software

– Implementing numerical solutions to DEs

(Euler/Runge Kutta) on a spreadsheet

– Using CAS to solve some DEs analytically

Differential equations in GeoGebra

Solving Differential Equations

Find a solution passing through some points other

than (1,1) such that y does not always decrease as

x increases.

(MEI A level FM Pure SAM)

Spreadsheet for the Euler method

FPT SAM question

Second Order Differential Equations

In the CAS view use:

SolveODE[ <Equation>, <Point(s) on f>, <Point(s) on f'> ]

e.g. SolveODE[y'' +2y' + y = sin(x), (0, 1), (0, 2)]

There are also a lot of good examples online, e.g.

• geogebra.org/m/zGah4NrW

• geogebra.org/m/y4Ercw2b

Or search for “Second Order ODE” at: geogebra.org/materials/

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MEI Conference 2017

Using technology to

investigate differential

equations

(FPT and more)

Tom Button tom.button@mei.org.uk

mei.org.uk/fpt

Using GeoGebra for Differential Equations

Slope fields and solving differential equations

To plot a slope field use SlopeField[f] in the Algebra View, e.g.

f(x,y)= 2x

SlopeField[f] To find a particular solution use SolveODE[f,A] in the Algebra View, e.g.

A=(1,0)

SolveODE[f,A] To find a general solution use SolveODE[equation] in the CAS View, e.g.

SolveODE[y’=2x]

Using spreadsheets for Numerical Methods

For the differential equation d𝑦

d𝑥= √𝑥𝑦

with initial conditions 𝑦 = 2 when 𝑥 = 2. Find an estimate for 𝑦(3) when

using a step size of ℎ = 0.1. Euler method:

𝑦𝑛+1 ≈ 𝑦𝑛 + ℎf(𝑥𝑛, 𝑦𝑛)

In the Algebra view: - f(x,y)=sqrt(x*y) - h=0.1

In the Spreadsheet view: - A1=”x”, B1=”y”, C1=”f” - A2=2, B2=2 - C2=f(A2,B2) - A3=A2+h - B3=B2+h*C2

Second Order Differential Equations

In the CAS view use: SolveODE[ <Equation>, <Point(s) on f>, <Point(s) on f'> ]

e.g. to solve d2𝑦

d𝑥2 + 2d𝑦

d𝑥+ 𝑦 = sin𝑥, 𝑦 = 1 and

d𝑦

d𝑥= 2 when 𝑥 = 0 :

SolveODE[y'' +2y' + y = sin(x), (0, 1), (0, 2)]