Matrix Theory: Analysing Coupled Linear Differential Equations
The Mathematical Set-Up and Story: Diagonalization, Eigenvalues and vectors, complex numbers and one-dimensional differential equations
A brief review of diagonalization, eigenvalues and eigenvectors, complex numbers, and one-dimensional differential equations towards a means for fully solving systems of differential equations.
Analysing Coupled Linear Differential Equations
Analysing solutions to systems of equations, understanding phase diagrams, and fully explaining long-term behaviour.
Prerequisite Knowledge: these videos presume familiarity with matrix diagonalisation, the theory of eigenvalues and eigenvectors of matrices, and the solution of single-variable linear differential equations from a first course in calculus (see the video course here: 
Matrices and their many Applications - James Tanton )
Practice Questions follow.
[Note: Site membership is required for the Exam Style Questions and an in-depth analysis/explanation of the applications to Lokta Volterra, Predator-Prey models follow at the bottom of this page]
Coupled DEs1: One Dimensional Differential Equations
See the structured, student notes sheet, designed to facilitate active listening, underneath the video below.. This is particularly effective when used in a mixed SL/HL classroom, or set as a flipped homework or online learning task.
Coupled DEs2: Two Dimensional Coupled Differential Equations
See the structured, student notes sheet, designed to facilitate active listening, underneath the video below.. This is particularly effective when used in a mixed SL/HL classroom, or set as a flipped homework or online learning task.
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Coupled DEs3: Analysing solutions
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Coupled DEs4: Complex Eigenvalues
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Optional extra: Where do Complex Numbers come from?
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below to view this video.
Optional extra: Discovering Euler's Trigonometric Formula
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Practice questions
Coupled Differential Equations - these questions offer practice at the concepts, techniques and ideas from this topic.
2nd order linear differential equation - an Introduction
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Student notes sheet1 (site subscription required for access)
Student's Notes sheet, to accompany the above One Dimensional Coupled Differential Equations video, with space for their working out.
Student notes sheet2
Student's Notes sheet, to accompany the above Two Dimensional Coupled Differential Equations video, with space for their working out.
Student notes sheet3
Student's Notes sheet, to accompany the above Coupled DEs4: Complex Eigenvalues video, with space for their working out.
IB exam style questions (site subscription required for access)
Coupled differential equations with Phase Portraits Exam Qu: (site membership is required to access these) use these Exam style questions to test if you feel ready, or what you need to review, ahead of moving on to the new topic or your end of unit test/mocks/final exams!
SOLUTIONS Coupled DEs and Phase Portraits
HL Couped DE, Phase Portraits ExamSet2 - a further 100 marks of exam style questions on this topic.
Teaching Activities (site subscription required for access)
Coupled Differential Equations and Phase Portraits - Predator-Prey Lotka Volterra Model
This resource provides an excellent, detailed analysis, history and explanation of the application of Coupled DEs and Phase Portraits to the Lotka Volterra, Predator-Prey models (that leads into plenty of IA possibilities). [Note: Site membership is required to access this resource].
Student notes sheet - 2nd order differential equation (site subscription required for access)
Student's Notes sheet, to accompany the above 2nd order linear differential equation - an Introduction video, with space for their working out.
IB exam style questions (site subscription required for access)
Second order linear differential equations: (site membership is required to access these) use these Exam style questions to test if you feel ready, or what you need to review, ahead of moving on to the new topic or your end of unit test/mocks/final exams!