Differential Equations

Overview

To provide students with knowledge of ordinary differential equations required for practising scientist and engineer.

Unit learning outcomes

Students who successfully complete this unit will be able to:

1. Systematically classify ordinary differential equations as linear or nonlinear, homogeneous or non-homogeneous, determine their order and degree and identify and solve Logistic, Bernoulli's and Riccati's equations
2. Analyse and solve second order ODEs with non-constant coefficients including Legendre equation using power series method
3. Apply Frobenius methods for solving equations with singular coefficients and Bessel functions
4. Generate orthogonal solutions to second order differential equations and use them as the basis for orthogonal eigenfunction expansion
5. Identify basic linear second order PDEs with constant coefficients: Laplace's, Poisson's, heat and wave equations, and solve them by separation of variables
6. Discuss physical examples requiring formulation of second order linear PDEs in simple rectangular and circular geometries: oscillating membranes; propagating waves, heat propagation

Teaching methods

Hawthorn

Assessment

(i) an aggregate mark of 50% or more, and(ii) at least 40% in the final exam.Students who do not successfully achieve hurdle requirement (ii) will receive a maximum of 45% as the total mark for the unit.

Content

• Review of simple ODEs: terminology, types of ODEs and review of methods of their solution studied previously
• Special first order ODEs: Verhulst (logistic) equation with applications, Bernoulli equation, Riccati equation.
• Second order ODEs with non-constant coefficients: variation of parameters, power series method, Legendre’s equation and Legendre polynomials, Frobenius method, Bessel equation and its solutions, Sturm-Liouville problems and orthogonal functions, orthogonal eigenfunction expansions.
• Linear second order partial differential equations: wave, heat and Laplace’s equations and their physical origins.
• Linear second order PDEs in multiple spatial dimensions: waves on a rectangular and circular membranes with applications, Laplacian in cylindrical and spherical co-ordinates, applications in complex geometries.