## MAS222 Differential Equations

 Both semesters, 2021/22 20 Credits Lecturer: Prof Elizabeth Winstanley Timetable Reading List Aims Outcomes Teaching Methods Assessment Full Syllabus

Differential equations arise in many areas within the physical, natural, and social sciences. They can be used to model a whole host of phenomena, from sub-atomic particles to competing animal species, fluid flow to stock markets. This module develops a core set of advanced mathematical techniques essential to the study of applied differential equations. Topics include the qualitative analysis of ordinary differential equations, the solution of second order linear ordinary differential equations with variable coefficients, first and second order partial differential equations, the method of characteristics and the method of separation of variables.

Prerequisites: MAS110 (Mathematics Core I); MAS111 (Mathematics Core II)
Corequisites: MAS211 (Advanced Calculus and Linear Algebra)

The following modules have this module as a prerequisite:

 MAS212 Scientific Computing and Simulation MAS316 Mathematical modelling of natural systems MAS320 Fluid Mechanics I MAS377 Mathematical Biology MAS414 Mathematical Modelling of Natural Systems MAS422 Magnetohydrodynamics

## Outline syllabus

• Qualitative analysis of 1D and 2D systems of ordinary differential equations (ODEs), including the classification of equilibrium points in 1D and 2D systems of ODEs
• Exact analysis of second order linear ODEs: normal form, reduction of order, and power series solutions
• Sturm-Liouville problems
• Second order partial differential equations (PDEs), Laplace’s equation, wave equation, heat equation, separation of variables.
• First order PDEs, method of characteristics.
• Second order hyperbolic PDEs, method of characteristics

## Aims

• To give an introduction to qualitative analysis of ordinary differential equations.
• To learn some basic methods for solving second order ordinary differential equations with variable coefficients.
• To give an introduction to the method of characteristics for first order linear partial differential equations.
• To give an introduction to the method of separation of variables for second order linear partial differential equations.
• To give an introduction to the properties of the solutions of the heat equation, wave equation and Laplace’s equation.

## Learning outcomes

• To be able to characterise systems of ordinary differential equation qualitatively.
• To be able to solve second order ordinary differential equations with variable coefficients using various techniques
• To be able to solve first order partial differential equations using the method of characteristics
• To be able to solve second order partial differential equations using the method of separation of variables and the method of characterisics

## Teaching methods

Lectures, tutorials, problem solving

40 lectures, 10 tutorials

## Assessment

One 1 hour exam on computer at the end of Semester 1 (10%)
One 2.5 hour written examination at the end of Semester 2 (90%)

## Full syllabus

Semester 1

• Revision of ordinary differential equations (ODEs)
• Qualitative analysis of first order ODEs: direction fields, autonomous equations, equilibrium points, phase lines.
• Planar first order autonomous systems: equilibrium points, trajectories, nullclines, classification of equilibrium points for linear systems, phase portraits in the neighborhood the equilibrium points
• Linearisation of nonlinear planar system
• Stability of equlibrium points: linear systems, effects of nonlinear terms
• Phase portraits of planar first order autonomous systems
• Second order linear ordinary differential equations with variable coefficients: boundary value problems, normal form, reduction of order
• Power series solution and Frobenius series solution: ordinary points and singular points
Semester 2
• Introduction and basic definitions
• Separation of variables for homogeneous problems: the heat equation, wave equation, Laplace's equation
• Separation of variables for inhomogeneous problems: the heat equation, wave equation Laplace's equation
• Method of characteristics for first order PDEs
• Method of characteristics for second order hyperbolic PDEs
• D'Alembert's solution of the one-dimensional wave equation.