
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 matrixexponentials 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.


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, food webs, to infectious diseases and neuronal oscillators. The focus of this course is on finding solutions analytically where possible, and else to characterize their longterm 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 the existence and uniqueness of solutions for a large class of equations. We then generalize these methods using the matrix exponential to cover ndimensional 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 deriving explicit solutions, we characterize their longterm 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 twodimensional systems using nullclines and the limiting behaviour of solutions using the PoincarĂ©Bendixson Theorem or Lasalle's Invariance Theorem.




 Assumed previous knowledgeKnowledge of calculus and linear algebra, in particular eigenvalues and eigenvectors. For nonAM students, the mathematics learning line suffices. 
Bachelor Applied Mathematics 
  Required materialsBook“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 secondhand) 

 Recommended materialsInstructional modesAssignmentPresence duty   Yes 
 Lecture
 Seminar
 Tutorial

 TestsTest ODE
 Challenges


 