MAU22205 Ordinary differential equations

Assessment Details

• This module is examined in a 2-hour examination at the end of Semester 1.
• Continuous assessment contributes 20% towards the overall mark.
• The module is passed if the overall mark for the module is 40% or more. If the overall mark for the module is less than 40% and there is no possibility of compensation, the module will be reassessed as follows: 
  1) A failed exam in combination with passed continuous assessment will be reassessed by an exam in the supplemental session; 
  2) The combination of a failed exam and failed continuous assessment is reassessed by the supplemental exam; 
  3) A failed continuous assessment in combination with a passed exam will be reassessed by one or more summer assignments in advance of the supplemental session.

Contact Hours

11 weeks of teaching with 3 lectures per week.

Learning Outcomes

On successful completion of this module, students will be able to

• Apply various standard methods (separation of variables, integrating factors, reduction of order, undetermined coefficients) to solve certain types of differential equations (separable, first-order linear, linear with constant coefficients).
• Give examples of differential equations for which either existence or uniqueness of solutions fails.
• Compute the exponential of a square matrix.
• Use either linearization or the Lyapunov theorems to check the stability of critical points for a given autonomous system.

Module Content

• Terminology (order, scalar vs. system, linear vs. nonlinear, invariant).
• Separable equations, first-order equations, Gronwall inequality.
• Existence and uniqueness of solutions, blow up in finite time.
• First-order linear systems, exponential of a square matrix.
• Reduction of order, undetermined coefficients, variation of parameters.
• Autonomous systems, stability theory, Lyapunov functions.

Recommended Reading

• The qualitative theory of ordinary differential equations by Brauer and Nohel.