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Module MA2327: Ordinary Differential Equations

Credit weighting (ECTS)
5 credits
Semester/term taught
Michaelmas term 2017-18
Contact Hours
11 weeks, 3 lectures including tutorials per week
Lecturer
Prof Paschalis Karageorgis
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, substitutions, undetermined coefficients) to solve various types of ordinary differential equations (separable, first-order linear, homogeneous, linear with constant coefficients);
  • provide examples of initial value problems to illustrate that such a problem may have infinitely many solutions or no solutions at all;
  • use the eigenvector method and/or the matrix exponential to solve systems of linear equations with constant coefficients;
  • prove various properties satisfied by fundamental matrices of linear systems and find such a matrix explicitly in a few special cases;
  • apply standard methods (Jacobian matrix, Lyapunov theorems) to check the critical points of an autonomous system for stability.
Module Content

The main concepts to be introduced in this module are the following.

  • General theory: order of an equation, direction field, initial value problem, existence and uniqueness, blow up, continuous dependence on initial data, separable equation, linear equation, integrating factor, homogeneous equation, Bernoulli equation, Gronwall inequality.
  • Linear systems: homogeneous system, superposition principle, linear independence of functions, eigenvector method, matrix exponential, fundamental matrix, variation of parameters, characteristic equation, method of undetermined coefficients, Wronskian, reduction of order.
  • Stability theory: autonomous system, equilibrium or critical point, stability, asymptotic stability, Jacobian matrix, Lyapunov function.
Module Prerequisites
MA1126 and MA1212.
Assessment
This module will be examined in a 2-hour examination in the Trinity term. Continuous assessment will count for 20% and the annual exam will count for 80%. Students who are required to take a supplemental exam will be assessed based on that exam alone.