Department of Mathematical Sciences



Differential Equations and Linear Algebra

The first part of the module is on ordinary differential equations and the theory of initial value problems. Several techniques for solving first order differential equations are introduced. This is followed by the solution of higher order linear differential equations, including methods of undetermined coefficients and variation of parameters. In conclusion of this part of the module the Laplace transform is introduced and some applications are discussed. The second part of the module follows on the matrix algebra which was studied in the first year course, and covers linear independence, eigenvalues and eigenvectors of matrices, orthogonal matrices and diagonalization.  Some applications of diagonalization are also discussed.

Module Information

  • 38571 214 (15) Engineering Mathematics 214
  • Second year, first semester of the Programs in Civil, Electrical, Industrial, Process and Mechanical Engineering.
  • Four lectures and one tutorial of 2 hours per week.
  • Prerequisite pass modules (PP³50): Engineering Mathematics 115 or Engineering Mathematics 145; Prerequisite module (KP³40): Engineering Mathematics 145.
  • Language specification : A & E


  • Prof A Fransman (Afrikaans) Groups 1 & 2.  Office 3006, Mathematical Sciences and Industrial Psychology building (e-mail:
  • Prof Z Janelidze (English) Group 3.  Office 1016B, Mathematical Sciences and Industrial Psychology building (e-mail:
  • Dr A Keet (English) Group 4.  Office 1010A, Mathematical Sciences and Industrial Psychology building (e-mail:

Learning Material

  • Prescribed textbook: DG Zill & WS Wright:  Advanced Engineering Mathematics (5th Edition), Jones and Bartlett Publishers, 2014.

Module Content

  • Differential equations: Selected material form Chapters 2 and 3 (2.4; 2.5; 3.1-3.6).
  • Laplace Transforms: Selected material from Chapter 4 (4.1; 4.2).
  • Linear Algebra: Vector spaces (Chapter 7.6).
  • Eigenvalues and diagonalisation:  Selected material from Chapter 8 (8.8; 8.10; 8.12).

Learning Opportunities

  • The course material is dealt with in detail during the lectures. Tutorials provide the opportunity to work on problems and discuss coursework under supervision of the lecturer.


  • This module uses the “Flexible Assessment” method, as applied in the Engineering Faculty.  For details, please refer to the Faculty’s Assessment Rules, which is available on the Moodle pages offered by the Engineering Faculty (in block titled “General Programme Information” on the left-hand side of the screen, just below the “Navigation” block).
  • The pass requirement for this module is a final mark FM ≥ 50%.
  • A semester mark (SM) is required by writing tutorial tests during weekly tutorials.
  • There are three formal scheduled assessment opportunities:  A1 (mid-semester test), A2 (first examination), and A3 (second examination).
  • The following weighting factors are used to determine the FM: WSM=0.1; WA1=0.4; WA2=0.5
  • The date and time of the semester test are determined by the Engineering Faculty ( and those of A2 and A3 are published on the University’s website.  For any further information on examinations and promotion, see the University Calendar, Parts 1 and 11.

General Information

  • Please note the following regarding the tutorials: (a) Attendance of all tutorials is compulsory. (b) No other appointments (academic or otherwise) may be observed during tutorial periods.
  • The solutions of tutorial problems and tutorial tests will be made available on web studies.

Study Hints

  • It is important that you understand the basic concepts very well, so that you are able to apply the theory.
  • To determine whether you have mastered the course material, you must work a variety of problems from the Exercises in the textbook.
  • You are welcome to consult your lecturer if there is anything which you do not understand, or if you get stuck with a problem. The lecturer will be available right after each lecture, as well as in his/her office (preferably by appointment).
  • Revise each section completely as soon as it has been discussed in class. It is very important that you do not fall behind, since it is very difficult to catch up.


The module is presented as part of all Programs in Engineering and offers basic training in Mathematics which is necesary for the successful completion of other modules within these Programs. The module supports the Program outcome that graduates should be able to use their knowledge of mathematics to solve engineering problems.

Learning Outcomes

After completing this course, students should be able to:

  • Solve first-order differential equations of several types; know the criteria for existence and uniqueness of a first-order differential equation.
  • Solve higher order linear differential equations, both homogeneous and non-homogeneous types. Know the criteria for the existence and uniqueness of higher order linear differential equations.
  • Solve initial value problems using Laplace transforms.
  • Understand the concept of linear independence and its application in the solution of systems of linear equations.
  • Calculate eigenvalues and eigenvectors of matrices; know the basic facts of diagonalization of matrices, and be able to apply these.