# ORDINARY DIFFERENTIAL EQUATIONS - 2024/5

Module code: MAT2007

## Module Overview

This module builds on the differential equation aspects of the Level 4 modules Calculus and Linear Algebra and considers qualitative and quantitative aspects of Ordinary Differential Equations.

### Module provider

Mathematics & Physics

TRONCI Cesare (Maths & Phys)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 57

Lecture Hours: 34

Tutorial Hours: 10

Guided Learning: 15

Captured Content: 34

Semester 1

None.

## Module content

Indicative content includes:

• Scalar first-order differential equations; review of separable and linear equations;

• Phase portraits on the line; equilibria and their stability;

• Theorems on existence, uniqueness, continuous dependence on initial conditions;

• Linear systems of differential equations: the solution set, solution matrix and Wronskian;

• Scalar, linear higher order differential equations: relation with systems of differential equations;

• Linear, autonomous systems of differential equations: relation between stability and eigenvalues; classification of planar phase portraits;

• Nonlinear systems: equilibria and their classification, linear stability analysis, Lyapunov functions, phase portrait near an equilibrium;

• If time allows: If time allows: gradient flows, saddle-node bifurcations, Hopf bifurcation.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-Semester Test (50 minutes) 20
Examination End-of-Semester Examination (2 hours) 80

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• Understanding of and ability to interpret and solve ODEs.

• Subject knowledge through the recall of key definitions, theorems and their proofs.

• Analytical ability through the solution of unseen problems in the test and examination.

Thus, the summative assessment for this module consists of:

• One in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcomes 1 and 2.

• A synoptic examination (2 hours), worth 80% of the module mark, corresponding to Learning Outcomes 1 to 3.

Formative assessment
There are two formative unassessed courseworks over an 11 week period, designed to consolidate student learning.

Feedback
Students receive individual written feedback on the formative unassessed coursework and the in-semester test. The feedback is timed so that feedback from the first unassessed coursework assists students with preparation for the in-semester test. The feedback from both unassessed courseworks and the in-semester test assists students with preparation for the end-of-semester examination. This written feedback is complemented by verbal and written feedback given in tutorials. Students also receive verbal and written feedback in office hours.

## Module aims

• This module aims to study both qualitative and quantitative aspects of Ordinary Differential Equations.

## Learning outcomes

 Attributes Developed 001 Students will be able to find exact solutions to certain types of differential equations. KCT 002 Students will be able to plot and interpret phase portraits on the line or in the plane. KCT 003 Students will be able to determine the stability of equilibria and periodic solutions. KCT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

• A detailed introduction to the theory of ordinary differential equations (ODEs).

• Experience (through demonstration) of the methods used to interpret, understand and solve ODEs.

The learning and teaching methods include:

• Three one-hour lectures per week for eleven weeks, with typeset notes to complement the lectures. The lectures provide a structured learning environment with opportunities for students to ask questions and to practice methods taught.

• There are two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive individual written feedback on these as guidance on their progress and understanding.

• Weekly tutorials for guided discussion of questions in problem sheets provided to and worked on by students in advance.

• Quizzes to recap material and provide Question and Answer opportunities for students.

• Lectures may be recorded. Lecture recordings are intended to give students the opportunity to review parts of the session that they might not have understood fully and should not be seen as an alternative to attendance at lectures.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: MAT2007

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourcefulness and Resilience and Sustainability. This module is designed to allow students to develop knowledge, skills, and capabilities in the following areas:

Digital Capabilities: Students have the opportunity to construct phase portraits encountered during the module via software (in particular, the pplane app or similar online tools), thus enhancing their digital experience.

Employability: The proficiency that students gain in solving dynamic system problems enhances analytical and problem-solving skills, which are widely valued by employers. The ability to apply differential equations to real-world challenges enhances their overall employability.

Global and Cultural Capabilities: Students enrolled in MAT2007 originate from various countries and possess a wide range of cultural backgrounds. During tutorials, student engagement in discussions naturally cultivates the sharing of different cultures.

Resourcefulness and Resilience: Solving complex problems with differential equations fosters student resourcefulness, encouraging creative problem-solving. This skillset builds resilience as students navigate the uncertainties of real-world applications.

Sustainability: Students are shown that ordinary differential equations can be used to model and analyse dynamic systems such those involved in environmental processes, resource management, and population dynamics. This mathematical foundation enables informed decision-making, emphasising the role of differential equations in addressing complex challenges for a more sustainable future.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics with Statistics BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics MMath 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics with Music BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Financial Mathematics BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MPhys 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MMath 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Economics and Mathematics BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2024/5 academic year.