 # MATH2121 Theory and Applications of Differential Equations

MATH2121 is a Mathematics Level II course. See the course overview below.

Units of credit: 6

Prerequisites: MATH1231 or MATH1241 or MATH1251 or DPST1014

Exclusions: MATH2018, MATH2019, MATH2221.

Cycle of offering: Term 2

Graduate attributes: The course will enhance your research, inquiry and analytical thinking abilities.

More information: This course outline (pdf) contains information about course objectives, assessment, course materials and the syllabus.

The Online Handbook entry contains up-to-date timetabling information.

The higher version of this course, MATH2221 Higher Theory and Applications of Differential Equations, is also offered yearly in Semester 2.

MATH2121 (alternatively MATH2221) is a compulsory course for Mathematics majors.

If you are currently enrolled in MATH2121, you can log into UNSW Moodle this course.

#### Course Aims

Differential Equations (DEs) were first introduced by Newton to describe the behaviour of dynamical systems such as the motions of the planets and the trajectories of cannon balls. DEs now have applications in fields as diverse as biology (spread of epidemics), medicine (growth of tumours), sociology (emigration rates), psychology (learning theories), economics (option pricing), chemistry (reaction rates), physics (dynamics of a laser) and engineering (electric circuits). Exact and numerical methods for solving DEs are therefore of fundamental importance for understanding nature and technology.

In first year, we learnt how to solve first order ordinary differential equations and second order ordinary differential equations with constant coefficients. In this course, we learn how to deal with second order ordinary differential equations with variable coefficients and give an introduction to partial differential equations. We also apply a dynamical systems approach to systems of ordinary differential equations and give an introduction to solving ordinary differential equations numerically. We learn how to find solutions of differential equations that obey prescribed boundary conditions. Not all differential equations can be solved in terms of "elementary" functions such as polynomials, exponentials or trigonometric functions. A major aim of this course is to teach you how to get information about the solution in these cases using power series methods and Frobenius' method. A second major aim is to learn how to find solutions to boundary value problems using Sturm-Liouville methods and Fourier series methods.

#### Course Description

The following topics are treated both theoretically and with illustrative applications in physics, engineering and biology.

Ordinary differential equations: first order, linear second order, variation of parameters, dynamical systems, power series representations and Frobenius method, orthogonal functions and Fourier series, initial and boundary value problems, eigenfunction expansions, Bessel's equation.

Partial differential equations: classification, method of separation of variables, application of Fourier series, heat equation, wave equation, Laplace's equation, applications of Bessel functions.