Using technology to investigate differential equations · Using GeoGebra for Differential Equations...
Transcript of Using technology to investigate differential equations · Using GeoGebra for Differential Equations...
Using technology
to investigate
differential
equations
(FPT and more)
Tom Button
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 [email protected]
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)]