MM507: Differential Equations
Comment
Entry requirements
Academic preconditions
Students taking the course are expected to:
- Have knowledge of how to implement algorithms as computer programs and compute numerical approximations to mathematical problems that don't allow a closed form solution.
- Be familiar with: systems of linear equations, matrices, determinants, vector spaces, scalar product and orthogonality, linear transformations, eigenvectors and eigenvalues, diagonalisation, polynomials, the concept of a function, real and complex numbers, differentiation and integration of functions of one and several variables, vector calculus.
Course introduction
The course builds on the knowledge acquired in the courses MM536 (Calculus for Mathematics), MM533 (Mathematical and Numerical Analysis) and MM538 (Algebra and Linear Algebra). The course gives an introduction to the treatment of ordinary differential equations and so provides the basis for taking further courses dealing with differential equations such as MM531, MM8AA and MM546.
The course is of high multidisciplinary value and gives an academic basis for a Bachelor Project in several core areas of Natural Sciences.
In relation to the competence profile of the degree it is the explicit focus of the course to:
- Give the competence to handle complex and development-oriented situations in study and work contexts.
- Give skills to:
- apply the thinking and terminology from the subject's basic disciplines.
- analyze and evaluate the theoretical and practical problems for the application of a suitable mathematical model.
- Give knowledge and understanding of:
- basic knowledge generation, theory and methods in mathematics.
- how to conduct analyses using mathematical methods and critically evaluate scientific theories and models.
Expected learning outcome
- formulate a differential equation as a model for a simple problem
- solve differential equations by methods taught in the course
- find steady states and analyse the asymptotic behaviour of simple systems of differential equations
- apply these results to examples
- formulate and present definitions, proofs and computations in a mathematically rigorous way
Content
1.1. First order differential equations and mathematical models.
1.2. Slope fields and initial value problems.
1.3. Euler's approximation.
1.4. Existence and uniqueness, Picard-Lindelöf theorem (as application of fixed point theorem).
1.5. Gronwall's Lemma and the convergence of Euler's method.
1.6. Analytic tools: integrating factors, separation of variables, and exact equations.
2.1. Systems of first order linear differential equations, and linear higher order differential equations: fundamental solutions, the solution space.
2.2. The Wronskian, Abel's theorem.
2.3. Analytic tools: undetermined coefficients and the variation of parameters.
Literature
Examination regulations
Exam element a)
Timing
Tests
Mandatory assignments
EKA
Assessment
Grading
Identification
Language
Examination aids
ECTS value
Indicative number of lessons
Teaching Method
In order to enable the students to achieve the learning objectives for the course, the teaching is organized so that there are 42 lecture hours, class lectures, etc. in one semester.
These teaching activities result in an estimated indicative distribution of the work effort of an average student in the following way:
- Intro phase: 28 hours
- Training phase: 14 hours
Activities during the study phase:
- preparation of exercises in study groups
- preparation of projects
- contributing to online learning activities related to the course
Teacher responsible
Additional teachers
| Name | Department | City | |
|---|---|---|---|
| Alexei Latyntsev | latyntsev@imada.sdu.dk | Institut for Matematik og Datalogi | |
| Jens Kaad | kaad@imada.sdu.dk | Institut for Matematik og Datalogi |