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Graduate courses

Departments' graduate courses for PhD-students.


Syllabus for

Academic year
TMV151 - Integral calculus and ordinary differential equations
Integralkalkyl och ordinära differentialekvationer
Syllabus adopted 2020-02-10 by Head of Programme (or corresponding)
Owner: TKMAS
7,5 Credits
Grading: TH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail
Education cycle: First-cycle
Major subject: Mathematics

Teaching language: Swedish
Application code: 55138
Open for exchange students: No
Only students with the course round in the programme plan

Module   Credit distribution   Examination dates
Sp1 Sp2 Sp3 Sp4 Summer course No Sp
0108 Examination 7,5c Grading: TH   7,5c   14 Jan 2021 pm J   07 Apr 2021 pm J,  16 Aug 2021 am J

In programs



Axel Målqvist

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General entry requirements for bachelor's level (first cycle)
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Specific entry requirements

The same as for the programme that owns the course.
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Course specific prerequisites

Introductary course in mathematics and programming in Matlab.


The purpose of the course is to, together with the other math courses in the program, provide a general knowledge in the mathematics required in further studies as well as in the future professional career.

Learning outcomes (after completion of the course the student should be able to)

  • define and compute upper and lower Riemann sums.
  • define the concept of the integral and explain the connection between the derivative and the integral.
  • define and compute generalized integrals.
  • apply and motivate methods for computing integrals, both analytically and numerically.
  •  explain the meaning of an ordinary differential eqation (ODE) and determine the order of the equation and whether its linear or non-linear.
  • determine if an ODE of first order has unique solution. 
  • apply and motivate analytical methods for solving ordinary differential equations.
  • derive Laplace transforms and use the transform to solve ODEs. 
  • define matrix and vector and their algebraic properties.
  • reformulate higher order ODEs as a system of first order ODEs.
  • define elementary functions as solutions to ODEs.
  • implement numerical methods to solve ODEs and determine which method is appropriate for which problem.
  • determine convergence order and stability region for numerical methods.
  • combine knowledge of different concepts in practical problem solving.


The two main themes of the course are the integral and ordinary differential equations. Equal emphasis is put on the three pillars mathematical theory, analytical techniques and numerical methods.

Integral: Antiderivatives and integrals, methods of integration, integration of rational functions, generalized integrals, numerical integration. Applications of the integral: area, volume, center of gravity, arc length, area and volume of solids of rotation.

Ordinary differential equations:  general non-linear frirst order system, existence and uniqueness of solution, analytical techniques for separable and linear ODEs, second order linear ODEs with constant coefficients, Laplace transform, matrices and vectors algebraic properties, system of ODEs and higher order ODEs as systems of first order, numerical methods for ODE, convergence and stability.


The material is presented in the form of lectures as well as classes and computer labwork in smaller groups.
Detailed information will be given on the course homepage at the start of the course.


S. Larsson, A. Logg, A. Målqvist, Analys och linjär algebra del II: Integralkalkyl och ordinära differentialekvationer 

Examination including compulsory elements

Detailed information concerning the examination will be given on the course homepage at the start of the course.
Examples on possible forms of examination include:
  • selected assignments may be presented orally or in writing during the course,
  • project work, individually or in groups,
  • written or oral exam during and/or after the course.
  • problems/assignments solved using the computer presented in writing and/or at the computer.

Page manager Published: Thu 04 Feb 2021.