The course Differential Equations deals with various analytical and numerical methods to study the properties of solutions to ordinary differential equations (ODEs) and partial differential equations (PDEs). More information on the aim and learning outcomes of the course can be found in the course plan (Links to an external site.).

Course responsible: Simone Calogero

Teaching assistant (exercise sessions): Anna Rohova

Student representatives: Ida Claesson, Aron Fredriksson

Literature

Simone Calogero: Ordinary and Partial Differential Equations: An Introduction.

• Version with hyperlinks suitable for screen reading (pdf). Requires Acrobate Reader (or similar)
• Version without hyperlinks suitable for printing (pdf)   

Remark: We skip the sections marked with (•)

Errata in the lecture notes (pdf) (updated January 8th)

Jupiter notebook with some relevant Python codes (ipynb)

Online resources (optional):

Several well written notes on ODEs and PDEs can be found here

Course discussion 

If you have a question about the course material you can ask it in the discussion thread at this link: discussion. However, the best way to ask questions is during the lectures and the exercise sessions. You may also use this forum to look for team mates for the project, if you are not already in a group.

Schedule

The course schedule can also be found on TimeEdit

Remarks:

• The exercise session is on Wednesdays afternoon, however some exercises are discussed in other lectures. Only the exercises listed in the schedule below are relevant for the exam. All solutions of these exercises can be found in the lecture notes.
• If a topic is marked as non-examinable it means that there will be no questions about this topic in the exam.

PART	1:	Ordinary differential equations
When	Where	Content
30/10 10-11.45	HA2	Notation for ODEs. Solvable ODEs. Integral factor method. Sections 1.1, 2.1, 2.2 (pages 13--15).
1/11 10-11.45	KC	Separation of variables method. Second order linear ODEs with constant coefficients.  Cauchy problem. Picard-Lindelöf's theorem.  Sections 2.2 (pages 15--19), 2.3.
1/11 13.15-15	KC	Exercises 1, 2, 3, 4, 6, 7, 9, 11, 13, 14, 23.
6/11 10-11.45	HA2	Global solvability of the Cauchy problem. The projectile model (non-examinable). Introduction to boundary value problems. Sections 2.4, 2.6, 2.7.
8/11 10-11.45	KB	Autonomous dynamical systems: Fixed points, linearization, stability. Speed of convergence to equilibrium  (non-examinable). Section 3.1.
8/11 13.15-15	KB	Exercises 21, 24 (skip this), 28, 31, 31, 32, 33, 34, 35, 36, 37.
10/11 10-11.45	KC	Phase portrait of dynamical systems. Hamiltonian systems. Section 3.2.
13/11 10-11.45	HA2	2-dimensional linear dynamical systems: Jordan form, solutions. Section 3.3 (pages 58--65).
15/11 10-11.45	HC2	Phase portrait of 2-dimensional linear dynamical systems. Node, focus, saddle, center. Section 3.3 (pages 65--73).
15/11 13.15-15	KB	Exercises 39, 40, 41, 42, 44, 46, 59, 60.
17/11 10-11.45	KA	Stability analysis of non-linear dynamical systems. Ljapunov functions (non-examinable). Section 3.5, 3.6.
20/11 10-11.45	EC	Numerical solutions of ODEs. Finite difference method. Convergence and stability. Implementation of the finite difference method with Python (non-examinable, but  important for the project). Sections 4.1, 4.2.
22/11 10-11.45	KC	ODE models. Lotka-Volterra system. The 2-body problem. All the material in this lecture is non-examinable, but is important for the project. Sections 5.1, 5.2.
22/11 13.15-15	KC	Exercises 54, 55, 56, 61, 62.
24/11 10-11.45	KA	Numerical solutions of boundary value problems for ODEs. Central difference method. Introduction to the finite element method.  All the material in this lecture is non-examinable. Section 4.3.
PART	2:	Partial differential equations
27/11 10-11.45	HA2	Introduction to PDEs. The linear transport equation.  Sections 1.2, 6.1.
29/11 10-11.45	KB	Conservation laws. Burgers equation. Breaking time.   Sections 6.2 (pages 135--139).
29/11 13.15-15	HC2	Exercises 84, 85, 87, 91, 92, 93, 94.
1/12 10-11.45	HC2	Weak solutions of conservation laws. Shock waves. Applications of conservation laws: the Euler equation (non-examinable).  Sections 6.2 (pages 139--142), 6.3.
4/12 10-11.45	HA2	Heat equation. General solution in the whole space. Applications to Brownian motion and heat flow (non-examinable). Sections 7.1, 7.4.
6/12 10-11.45	HC2	Initial boundary value problem for the heat equation. Fourier series solution in one dimension. Section 7.2.
6/12 13.15-15	KB	Exercises 95, 96, 97, 98, 99, 100, 101, 102, 106.
8/12 10-11.45	HC2	Finite difference method for the one dimensional heat equation (non-examinable). Section 7.3.
11/12 10-11.45	HA2	Wave equation. Conservation of energy. Uniqueness of solutions to the initial value problem. d'Alembert formula.   Section 8.1, 8.2.
13/12 10-11.45	KB	Initial boundary value problem for the one dimensional wave equation. Fourier series solution. Applications of the wave equation: sound and electromagnetic waves (non-examinable). Sections 8.3, 8.5.
13/12 13.15-15	KB	Exercises 110, 111, 112, 113, 114, 115, 132, 133, 134.
15/12 10-11.45	HC2	Boundary value problem for the Poisson equation. Weak (or variational) formulation. Finite element method in 2 dimensions (non-examinable). Sections 9.1, 9.2, 9.3.

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Assignments

The assignment consists of Project 2 (Lotka-Volterra system) or Project 3 (2-body problem) in the lecture notes. The choice of the project is left to you. Note that each project consists of 2 parts. The assignment gives max 5 points. 

Remarks:

• The assignment is not compulsary, but strongly recommended.
• The assignment must be submitted in a single Jupiter notebook at  this link: submission 
• The assignment can be worked out on groups of max 3 students. However, each member of the group has to submit the assignment. Write the name of the students you worked with in the comment box of the submission page . 
• The deadline for submission is December 31, 2023 at 23.59. The grade on the assignment (max 5 points) will be communicated in the first week of January. The grade points of the assignment are valid only for the exams in 2024. 
• The assignment should start with an Introduction outlining the content and purpose of the project. The Python codes should contain brief comments explaining what each main block of code is doing. You can write the assignment in Swedish or English.
• If you submitted the project last year, then you can resubmit the same project this year. However you have to be the single author of the assignment and you should address my comments from last year (if any) in the new submission. Write in the comment box of the submission that this is a re-submission from last year. 
• All assignments will be checked for plagiarism.

Information on the exam

The final written exam gives max 30 points. The following rule will be used to transform the exam + assignment points to the finale grade:

Exam + assignment points	Final grade
0 - 17	Not passed
18 - 22	3
23 - 27	4
28 - 35	5

The exam consists of two parts. In the first part you will be asked to give and explain two definitions from the following list (max 3+3 points):

ODEs: Definition 3.2, Definition 3.4, Definition 3.5, Definition 3.7, Definition 3.29, Definition 4.3

PDEs: Definition 6.1, Definition 6.2, Definition 6.5, Definition 7.2, Definition 8.1

The explanation of each definition is expected to be about 1 page. You should clarify the motivation behind the concept being defined and why it is useful. Use pictures and examples if you like. If the definition contains a mathematical formula, you should explain the meaning of all symbols in the formula. Here is an example on how to answer this question: pdf.

The second part of the exam consists of 6 exercises, each giving max 4 points. 1 exercise will be  theoretical, similar (perhaps even identical) to one of the exercises marked with the symbol (?) in the lecture notes (and listed in the course schedule), while the other 5 will be computational (e.g., derive the stability properties of fixed points, computing solutions of PDEs, etc.). 

Remarks: 

• The exam can be written in English or Swedish.  According to Chalmers rules, you can bring an English-Swedish dictionary at the exam. Digital dictionaries are not allowed.
• No aids are permitted for the exam (not even a calculator)
• The text of the exam will contain a table of standard integrals and some other formulas, which you may or may not need to solve some of the exercises. Other non-standard formulas, e.g., trigonometric or vector identities, will be provided if you need them. 
• As a general rule, if a formula appears in a definition or in the statement of a theorem (not in the proof!) in the lecture notes, then you have to remember it for the exam.

Solution of 2024 exams

January (pdf)   Exam review: January 26th, MVF31, h.12-13.

April (pdf)

Old Exams

2023: January (pdf), April (pdf), August (pdf)

Sample Exam (pdf) Solution (pdf)

Examination procedures

In Chalmers Student Portal (Links to an external site.) you can read about when exams are given and what rules apply to exams at Chalmers. In addition to that, there is a schedule (Links to an external site.) when exams are given for courses at the University of Gothenburg.

Before the exam, it is important that you sign up for the examination. You sign up through Ladok (Links to an external site.). 

At the exam, you should be able to show valid identification.

After the exam has been graded, you can see your results in Ladok.

At the annual (regular) examination: 
When it is practical, a separate review is arranged. The date of the review will be announced here on the course homepage. Anyone who can not participate in the review may thereafter retrieve and review their exam at the Mathematical Sciences Student office (Links to an external site.). Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

At re-examination: 
Exams are reviewed and retrieved at the Mathematical Sciences Student office (Links to an external site.). Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

Course evaluation

At the beginning of the course, at least two student representatives should have been appointed to carry out the course evaluation together with the teachers. The evaluation takes place through conversations between teachers and student representatives during the course and at a meeting after the end of the course when the survey result is discussed and a report is written. 

Guidelines for Course evaluation (Links to an external site.) in Chalmers student portal.

 

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