# ACM10060 Applications of Differential Equations

This course introduces students to the theory of differential equations and dynamical systems and to their many applications as mathematical models. The topics covered prepare the student for more advanced subjects in ordinary differential equations (ODEs), dynamical systems theory, numerical methods and partial differential equations.

Course Outline:

A) 1st-order ODEs:
Direction fields, equilibria, bifurcations.
Linear equations: Integrating factor method.
Separable equations.
Exact equations.

B) 2nd-order ODEs:
Definitions.
Linear vs. nonlinear.
Dimensional analysis.
Examples.

C) 2nd-order linear constant-coefficient ODEs:
Homogeneous equations: Characteristic equation method (real, complex and double roots).
Non-homogeneous equations: Method of undetermined coefficients.
Examples: Forced and damped systems, mechanical oscillations, resonances, etc.

D) Systems of 2 coupled ODEs:
Linear and nonlinear.
Phase plane analysis.
Critical points and classifications in terms of eigendirections and eigenvalues.
Time permitting: Jacobi last multiplier method (integrating factor).
Time permitting: Non-autonomous systems and chaos.
Examples: Population models, the tragedy of the commons, epidemic models, etc.

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Curricular information is subject to change

Learning Outcomes:

On completion of this module students should be able to:

1) Construct intermediate linear and nonlinear mathematical models, based on concepts such as dimensional analysis and the continuum hypothesis.

2) Solve differential equations analytically, using methods such as:
Partial fraction decomposition.
Separation of variables.
Chain rule.
Nonlinear mappings.
Characteristic equation method.
Integrating factor method.
Phase-plane analysis: Critical points; separatrices; linearisation near critical points.
Matrix methods.

3) Analyse properties of the solutions and describe the meaning of the solutions for the phenomena studied. Applications may include:
One-dimensional mechanical systems (linear and nonlinear).
The falling skydiver.
Nonlinear motion of a projectile.
Resonant systems with external forcing.
Nonlinear high-dimensional models such as the prey-predator model.
Population models: The effect of harvesting; the tragedy of the commons.
The famous Lorenz 3D atmospheric model leading to chaotic orbits.
The Brusselator and other chemical clocks.

Student Effort Hours:
Student Effort Type Hours
Specified Learning Activities

24

Autonomous Student Learning

40

Lectures

36

Tutorial

12

Total

112

Approaches to Teaching and Learning:
Lectures, tutorials, enquiry and problem-based learning, including WeBWorK automatic assessment.

These activities constitute the basis for the student’s learning, by engaging the student in actual hard work: listening, writing, studying, solving problems and discussing problems with peers, tutors and lecturer.

These activities are complemented by the availability of online lecture notes and the help from the Maths Support Centre.
Requirements, Exclusions and Recommendations

Not applicable to this module.

Module Requisites and Incompatibles
Incompatibles:
ACM10100 - Differential & Diff Equations, MST30040 - Differential Equations

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Examination: End of Trimester Exam 2 hour End of Trimester Exam No Standard conversion grade scale 40% No

60

Continuous Assessment: In class tests & WeBWorK automatic assignments Varies over the Trimester n/a Standard conversion grade scale 40% No

40

Carry forward of passed components
No

Resit In Terminal Exam
Autumn Yes - 2 Hour
Feedback Strategy/Strategies

• Group/class feedback, post-assessment
• Online automated feedback

How will my Feedback be Delivered?

Group/class feedback, post-assessment: This is implemented by the lecturer and tutors, who will go through solutions of selected problems. Online automated feedback: This is implemented in the context of WeBWorK automatic assessment (gradable), which tells the student whether their answers are correct or not.

Name Role
Tiziana Comito Tutor
Mr Kevin Cunningham Tutor
Giulio Taiocchi Tutor
Timetabling information is displayed only for guidance purposes, relates to the current Academic Year only and is subject to change.

Spring

Lecture Offering 1 Week(s) - 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33 Fri 14:00 - 14:50
Lecture Offering 1 Week(s) - 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33 Thurs 12:00 - 12:50
Lecture Offering 1 Week(s) - 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33 Tues 11:00 - 11:50
Tutorial Offering 1 Week(s) - 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33 Wed 12:00 - 12:50
Tutorial Offering 2 Week(s) - 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33 Wed 17:00 - 17:50
Tutorial Offering 3 Week(s) - 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33 Wed 13:00 - 13:50
Spring