MATH11222 - 2007/2008
General Information
- Title: Calculus and Applications
- Unit code: MATH10222
- Credits: 10
- Ths course unit consists of the first half of MATH10222. It may not be taken with that course unit.
- Prerequisites: MATH10121
- Co-requisite units: None
- School responsible: Mathematics
- Members of staff responsible: Prof. Matthias Heil
Specification
Aims
Many
Brief Description of the unit
The course is devoted to an introduction to the study of ordinary differential equations (ODEs).
Learning Outcomes
Future topics requiring this course unit
A large number of course units in applied mathematics, pure mathematics and statistics.
Syllabus
- General introduction. Notation. What are ODEs? Implicit versus explicit form. Classification: order, linearity, autonomous ODEs. Boundary and initial conditions. Boundary and initial value problems. Existence and uniqueness for linear and nonlinear ODEs. [3]
- First-order ODEs. Graphical methods; separable ODEs, ODEs of homogeneous type; integrating factor. [4]
- Second-order ODEs. Existence and uniqueness. Linear ODEs: superposition of solutions, fundamental solutions and the general solution for homogeneous ODEs. The general solution of constant-coefficient ODEs; particular solutions for specific RHS; the method of undetermined coefficients. [If time permits (probably not): Power series exapansions about regular points.] Some nonlinear ODEs with special properties (autonomous ODEs and ODEs that do not contain the dependent variable). [8]
- Mechanics applications of second-order ODEs Damped harmonic motions of mechanical oscillators; harmonic forcing and resonance. [3]
- Non-dimensionalisation and scaling. Exploiting small parameters in an ODE: perturbation methods. Motivation via the roots of quadratic polynomials (singular perturbations only mentioned); applications to selected (regularly perturbed) ODEs. [4]
Textbooks
Teaching and learning methods
Four lectures and one supervision class each week. In addition, students should expect to spend at least six hours each week on private study. The course unit only lasts for the first six weeks of the semester.