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Course module: 202001354
202001354
Ordinary Differential Equations
Course info
Course module202001354
Credits (ECTS)4
Course typeStudy Unit
Language of instructionEnglish
Contact persondr. H.G.E. Meijer
E-mailh.g.e.meijer@utwente.nl
Lecturer(s)
Lecturer
prof.dr. C. Brune
Lecturer
prof.dr.ir. B.J. Geurts
Examiner
dr. H.G.E. Meijer
Contactperson for the course
dr. H.G.E. Meijer
Academic year2020
Starting block
1B
Application procedureYou apply via OSIRIS Student
Registration using OSIRISYes
Aims
After following this course the student:
  • is able to check if a solution of an ordinary differential equation (ODE) exists, and whether it is unique;
  • can solve ODEs analytically using separation of variables, integrating factors or variation of constants;
  • is able to compute matrix-exponentials to solve linear inhomogeneous first order systems;
  • can determine equilibria and analyse their stability using linearisation or Lyapunov functions;
  • is able to classify solutions for planar systems using nullclines, polar coordinates, conserved quantities and the theorem of PoincarĂ©-Bendixson.
Content
In this course we study systems of ordinary differential equations (ODEs) in two or more dimensions. Such equations appear in many fields. Physical systems from classical mechanics range from the sling, coupled pendula to masses moving in a potential field. Models from biology range from preypredator systems, foodwebs, to infectious diseases and neuronal oscillators. The focus of this course is on finding solutions analytically where possible, and else to characterize their long-term behavior using geometric methods. After exploratory simulations, a typical problem would be to show that a model has a periodic orbit, or to determine which initial values approach a certain steady state. Throughout the course, applications serve as motivating examples.
A differential equation is a relation of a function and its derivatives. Normally an equation yields a (numerical) value as answer, but here we search for a function. We first recall solution methods for ordinary differential equations depending on a single variable, e.g. time, including separation of variables and variation of constants. We then prove existence and uniqueness of solutions for a large class of equations. We then generalize these methods using the matrix exponential to cover n-dimensional linear inhomogeneous systems, and discuss stability.
In the last part, we treat nonlinear systems of ODEs that can exhibit oscillatory and even chaotic dynamics. As the nonlinearities typically prohibit to derive explicit solutions, we characterize their long-term behaviour with other (geometric) methods such as polar coordinates, conserved quantities or Lyapunov functions. Using linearization we classify equilibria and their stability. In particular, we study two-dimensional systems using nullclines and the limiting behaviour of solutions using the Poincaré-Bendixson Theorem or Lasalle's Invariance Theorem.
 
Assumed previous knowledge
Knowledge of calculus and linear algebra, in particular eigenvalues and eigenvectors. For non-AM students, the mathematics learning line suffices.
Module
Module 6
Participating study
Bachelor Applied Mathematics
Required materials
Book
“Differential Equations with Boundary Value Problems” by JC Polking. ISBN: 9781292039152. (The old hardcover book (ISBN 9780134689500) can also be used, but is available only second-hand)
Recommended materials
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Instructional modes
Assignment
Presence dutyYes

Lecture

Seminar

Tutorial

Tests
Test ODE

Challenges

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Kies de Nederlandse taal