## MA202-5-SP-CO:

Ordinary Differential Equations

## Key module for

BSC N233DT Actuarial Science (Including Placement Year),

BSC N323 Actuarial Science,

BSC N323DF Actuarial Science,

BSC N323DT Actuarial Science,

BSC N324 Actuarial Science (Including Year Abroad),

BSC N325 Actuarial Science (Including Foundation Year),

BSC L1G2 Economics and Mathematics (Including Placement Year),

BSC LG11 Economics and Mathematics,

BSC LG18 Economics and Mathematics (Including Foundation Year),

BSC LG1C Economics and Mathematics (Including Year Abroad),

BSC L1G1 Economics with Mathematics,

BSC L1G3 Economics with Mathematics (Including Placement Year),

BSC L1G8 Economics with Mathematics (Including Foundation Year),

BSC L1GC Economics with Mathematics (Including Year Abroad),

BSC GN13 Finance and Mathematics,

BSC GN15 Finance and Mathematics (Including Placement Year),

BSC GN18 Finance and Mathematics (Including Foundation Year),

BSC GN1H Finance and Mathematics (Including Year Abroad),

BSC G100 Mathematics,

BSC G102 Mathematics (Including Year Abroad),

BSC G103 Mathematics (Including Placement Year),

BSC G104 Mathematics (Including Foundation Year),

MMATG198 Mathematics,

BSC 5B43 Statistics (Including Year Abroad),

BSC 9K12 Statistics,

BSC 9K13 Statistics (Including Placement Year),

BSC 9K18 Statistics (Including Foundation Year),

BSC G1G4 Mathematics with Computing (Including Year Abroad),

BSC G1G8 Mathematics with Computing (Including Foundation Year),

BSC G1GK Mathematics with Computing,

BSC G1IK Mathematics with Computing (Including Placement Year),

BSC G1F3 Mathematics with Physics,

BSC G1F4 Mathematics with Physics (Including Placement Year),

BSC G1F5 Mathematics with Physics (Including Foundation Year),

BSC GCF3 Mathematics with Physics (Including Year Abroad),

MSCIN399 Actuarial Science and Data Science,

MSCIG199 Mathematics and Data Science,

BSC N333 Actuarial Studies,

BSC N333DT Actuarial Studies,

BSC N334 Actuarial Studies (Including Placement Year),

BSC N334DT Actuarial Studies (Including Placement Year),

BSC N335 Actuarial Studies (Including Year Abroad)

## Module description

This module provides an overview of standard methods for solving single ordinary differential equations and systems of ordinary differential equations, with an introduction to the underlying theory. The first part is devoted to basic theory and methods for solving scalar ODEs. The second part of the module is devoted to the study of Systems of linear ODEs.

The subject of ordinary differential equations is a very important branch of Mathematical Analysis and has deep conections with Applied Mathematics. Many phenomena from Physics, Biology, Engineering, Chemistry, Finance, among others, may be described using ordinary differential equations. To understand the underlying processes, we have to find and interpret the solutions to these equations.

>## Module aims

The aims of this module are:

- To introduce students to the basic theory of ordinary differential equations.
- To give a competence in solving ordinary differential equations by using analytical methods.

## Module learning outcomes

By the end of the module, students will be expected to:

- Use some of the standard methods to solve first order ODEs;
- Use some of the standard methods to solve second order ODEs;
- Be familiar with the basic theory and be able to solve higher order linear ODEs;
- Be familiar with the basic theory and be able to solve systems of first order linear ODEs;
- Be aware of the implications of existence and uniqueness theorems.

## Module information

*Indicative syllabus:*

- Revision of known knowledge on ODEs: Classification of Differential Equations, First order differential equations: Linear Equations with Variable Coefficients, Separable Equations.
- First order differential equations: Differences Between Linear and Nonlinear Equations, Exact Equations and Integrating Factors, (Euler) homogeneous equations, Bernoulli equation, Riccati equation. Equilibrium solutions of autonomous ODEs. The Existence and Uniqueness Theorem.
- Second Order Linear Equations: Homogeneous Equations with Constant Coefficients, Fundamental Solutions of Linear Homogeneous Equations, Linear Independence and the Wronskian, Complex Roots of the Characteristic Equation, Repeated Roots; Reduction of Order; The Existence and Uniqueness Theorem.
- Second Order Linear Equations: ODEs with missing variables, Non-homogeneous Equations, Method of Undetermined Coefficients, Variation of Parameters.
- Higher Order Linear Equations: General Theory of nth Order Linear Equations, Homogeneous Equations with Constant Coefficients, The Method of Undetermined Coefficients, The Method of Variation of Parameters.
- Systems of First Order Linear Equations: Basic Theory of Systems of First Order Linear Equations including Linear Systems with Constant Coefficients. The fundamental matrix of a system. The fundamental matrix as matrix exponential.

## Learning and teaching methods

Teaching in the School will be delivered using a range of face-to-face lectures, classes, and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.

## Bibliography*

- William E. Boyce, Richard C. DiPrima and Douglas B. Meade (2017)
*Boyce's elementary differential equations and boundary value problems*. Eleventh edition, Global edition. Hoboken, NJ: Wiley. Available at: https://app.kortext.com/Shibboleth.sso/Login?entityID=https://idp0.essex.ac.uk/shibboleth&target=https://app.kortext.com/borrow/272233.

The library makes provision for all reading list items, with digital provision where possible, and these resources are shared between students.

Further reading can be obtained from this module's reading list.

## Assessment items, weightings and deadlines

Coursework / exam | Description | Deadline | Coursework weighting |
---|---|---|---|

Coursework | Written test 2 | ||

Coursework | Written test 1 | ||

Exam | Main exam: In-Person, Open Book (Restricted), 120 minutes during Summer (Main Period) | ||

Exam | Reassessment Main exam: In-Person, Open Book (Restricted), 120 minutes during September (Reassessment Period) |

### Exam format definitions

**Remote, open book:**Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.**In-person, open book:**Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.**In-person, open book (restricted):**The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.**In-person, closed book:**The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary, for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.

Your department will provide further guidance before your exams.

### Overall assessment

Coursework | Exam |
---|---|

20% | 80% |

### Reassessment

Coursework | Exam |
---|---|

20% | 80% |

## External examiner

**70**hours,

**66 (94.3%)**hours available to students:

**4**hours not recorded due to service coverage or fault;

**0**hours not recorded due to opt-out by lecturer(s).

*** Please note:** due to differing publication schedules, items marked with an asterisk (*) base their information upon the previous academic year.

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