✩ Language: English
✩ Place: Czech Technical University in Prague, Prague, Czech Republic
✩ Category: Theory + Labs
✩ Credits: 7 credits
✩ Classes: Tuesday 14.30 – 16.00 (Lecture) ★ Wednesday 9.15 – 10.45 (Lecture) ★ Wednesday 11.00 – 12.30 (Practice)
✩ When: Summer Term 2025 – 2026 (February 16 – May 24, 2026)
✩ Syllabus
✩ Evaluation system
★Lectures: attendance is not mandatory, but highly recommended.
★Practice: attendance is mandatory, 2 absences are tolerated.
★Midterm: it can yield up to 20 points, 10 is the minimum for passing it.
★Homeworks: 10 accepted homeworks is one of the requirements for zápočet (assessment).
★Zápočet (assesment): participation in labs + at least 10 points from the midterm + completing at least 10 homeworks.
★Final exam: it has 2 parts, a written test (mandatory) and an oral exam (optional). Only students with zápočet will be admitted to the final exam. Written test will consist of 4 problems for 20 points each, with 90 minutes allowed for solving them. Oral exam will look at theory, it can bring up to 10 points and allows students to improve their grade. It can be taken only by students who already passed the final based on their tests (see below).
★Grading: The grade is based on 3 inputs:
S is the points earned in the midterm decreased by 10, yielding a number from the range 0–10;
P is the points from the written test (range 0–80); and
U is the points from the (optional) oral exam (range 0–10).
The grade is determined by the following algorithm:
1) If a student did not score at least 40 pts on the final written test, the student failed the exam.
2) Assume that a student scored at least 40 pts on the final written test. If the sum P + S is not at least 50, the student failed the exam.
3) Assume that a student scored at least 40 pts on the final written test and also P + S is at least 50. Then the student passed the exam, we set U = 0 and the student can try to improve this number with the oral exam.
The grade is then determined by the total P + S + U according to the following key:
➤ 50 – 59 points: E
➤ 60 – 69 points: D
➤ 70 – 79 points: C
➤ 80 – 89 points: B
➤ 90 – 100 points: A
❖ You have at most 3 attempts to pass the final exam, each is entirely independent.
❖ Repeating the exam is obvious in case you fail, but you can also refuse a grade when you are not happy with it, then you are assigned the grade F and you can try again (if you still have attempts left).
❖ The midterm and the final test are subject to the following rules: No textbooks, no notes, no calculators, no cellphones are allowed.
❢❢ All the notes on this webpage are entirely based on Prof. Petr Habala’s classes and material.
★ Week 1 (February 17th and February 18th)
✩ Introduction to ODEs. Method of separation
✩ Some popular applications
Notes:
Slides: Week 1
***
★ Week 2 (February 24th and February 25th)
✩ Analyzing solutions
✩ Variation of parameter
Notes:
Slides: Week 2 (the same as last week)
***
★ Week 3 (March 3rd and March 4th)
✩ Introduction to Numerical Analysis. Erros and their propagation in calculations
✩ Numerical derivation
✩ Numerical integration
Notes:
Slides: Week 3 (errors in calculations and numerical derivation/integration)
***
★ Week 4 (March 10th and March 11th)
✩ Solving ODEs numerically
✩ The Euler method
✩ Global and local errors, and order of the method
Notes:
Slides: Week 4 (EDOs numerically and the Euler method)
***
★ Week 5 (March 17th and March 18th)
✩ Linear differential equations
✩ Homogeneous equations
✩ Asymptotic growth at infinity of solutions
Notes:
Slides: Week 5 (Linear different equations)
***
★ Week 6 (March 24th and March 25th)
✩ Structural theorems for non-homogeneous linear ODEs
✩ The method of undetermined coefficients for special RHS
✩ The superposition principle
✩ Method of variation for linear ODEs
Notes:
Slides: Week 6 (same as last week)
***
★ Week 7 (March 31st and April 1st)
✩ Solving equations numerically: roots
✩ The bisection, Newton and secant methods
✩ Stopping conditions and order of the method
Notes:
#13 March 31st
#14 April 1st
Slides: Week 7 (Roots of functions)
★ Week 1 – February 18th – Solving separable ODEs – Homework #1
★ Week 2 – February 25th – Slope field and variation of parameter – Homework #2
★ Week 3 – March 4th– Erros, derivation and integral numerically – Homework #3
★ Week 4 – March 11th – ODEs numerically and the Euler method – Homework #4
★ Week 5 – March 18th – Homogeneous linear ODEs – Homework #5
★ Week 6 – March 25th – Non-homogeneous linear ODEs – Homework #6
★ Week 7 – April 1st – The bisection and Newton methods – Homework #7
Week 8
Week 9
Week 10
Week 11 (midterm)
Week 12
Week 13
Week 14
★ (Week 1) Lecture #1: Tuesday, February 17th, 2026
– Introduction
– ODEs of order 1
– Solving ODEs using separation
★ (Week 1) Lecture #2: Wednesday, February 18th, 2026
– Two more examples using separation
– Applications of ODEs of order 1 – Exponential growth
– Applications of ODEs of order 1 – Newton’s law of cooling
– Existence of solutions: Peano and Picard theorems
★ (Week 1) Practice #1: Wednesday, February 18th, 2026
– Solving separable ODEs
★ (Week 2) Lecture #3: Tuesday, February 24th, 2026
– Analyzing solutions
– Slope field
– Stability
★ (Week 2) Lecture #4: Wednesday, February 25th, 2026
– Variation of parameter
– Applications of ODEs of order 1 – Logistic growth
– Applications of ODEs of order 1 – Free fall with air resistance
★ (Week 2) Practice #2: Wednesday, February 25th, 2026
– Slope field and variation of parameter
★ (Week 3) Lecture #5: Tuesday, March 3rd, 2026
– Introduction to Numerical Analysis
– Errors in calculations
– Floating point representation and error propagation
– The Taylor approximation and its error – the O-notation
★ (Week 3) Lecture #6: Wednesday, March 4th, 2026
– Numerical derivation
– Numerical integration
– The trapezoid method
– The Simpson method
★ (Week 3) Practice #3: Wednesday, March 4th, 2026
– Numerical Analysis: error, derivation and integration
★ (Week 4) Lecture #7: Tuesday, March 10th, 2026
– Solving ODEs numerically
– The Euler Method
– Global and local errors, and order of the method
– The integral approach for the Euler’s method
★ (Week 4) Lecture #8: Wednesday, March 11th, 2026
– The Heun method
– The midpoint method (RK2)
– Runge-Kutta methods
– Estimating error via control method
★ (Week 4) Practice #4: Wednesday, March 11th, 2026
– The Euler method and predicting error using order
★ (Week 5) Lecture #9: Tuesday, March 17th, 2026
– Linear differential equations
– Structural theorems
– Homogeneous equations
★ (Week 5) Lecture #10: Wednesday, March 18th, 2026
– Complex characteristic numbers
– Asymptotic rate of growth at infinity of solutions
– An example of a homogeneous equation with a parameter
★ (Week 5) Practice #5: Wednesday, March 18th, 2026
– Linear ODEs – homogeneous case
★ (Week 6) Lecture #11: Tuesday, March 24th, 2026
– Structural theorems for non-homogeneous linear ODEs
– The method of undetermined coefficients for special RHS
– The superposition principle
★ (Week 6) Lecture #12: Wednesday, March 25th, 2026
– Method of variation for linear ODEs
– An example without constant coefficients
★ (Week 6) Practice #6: Wednesday, March 25th, 2026
– Solving general linear ODEs (undetermined coefficients)
★ (Week 7) Lecture #13: Tuesday, March 31st, 2026
– Finding roots of equations numerically
– The bisection method
– The Newton method
– Stopping conditions
– Speed of convergence
★ (Week 7) Lecture #14: Wednesday, April 1st, 2026
– The secant method
– A summary of all three methods and comparison between them
– The Babylonian method
– The Newton-Raphson method
★ (Week 7) Practice #7: Wednesday, April 1st, 2026
– Finding roots/solving equations numerically (classical methods)
★ (Week 8) Lecture #15: Tuesday, April 7th, 2026
– The fixed-point method
– Convergence results
– Cobweb pattern and the fixed-point method
★ (Week 8) Lecture #16: Wednesday, April 8th, 2026
– Relaxation
– The optimal in the relaxation procedure
– Root finding through fixed points (relaxation for root problems)
– Review of analytic methods: recognizing the appropriate technique
– Review of eigenvalues and eigenvectors
★ (Week 8) Practice #8: Wednesday, April 8th, 2026
– Fixed point iteration
★ (Week 9) Lecture #17: Tuesday, April 14th, 2026
– Systems of linear ODEs of order 1
– Introduction and general theory
– Matrix setup for systems
– Homogeneous systems
★ (Week 9) Lecture #18: Wednesday, April 15th, 2026
– How to find a fundamental matrix
– Non-homogeneous systems of equations
– The variant method
– The row variation
★ (Week 9) Practice #9: Wednesday, April 15th, 2026
– Solving homogeneous systems of ODE
★ (Week 10) Lecture #19: Tuesday, April 21st, 2026
– Midterm (50 minutes to 80 minutes)
★ (Week 10) Lecture #20: Wednesday, April 22nd, 2026
– Stability of solutions
– Solving systems of 1st order ODEs numerically
★ (Week 10) Practice #10: Wednesday, April 22nd, 2026
– Solving non-homogeneous systems of linear ODEs
★ (Week 11) Lecture #21: Tuesday, April 28th, 2026
– Systems of linear equations
– Gaussian elimination method: error and numerical stability
– Gaussian elimination method: computational complexity
★ (Week 11) Lecture #22: Wednesday, April 29th, 2026
– The Gauss-Jordan elimination method
– The back substitution method
★ (Week 11) Practice #11: Wednesday, April 29th, 2026
– Solving systems of linear equations by elimination
★ (Week 12) Lecture #23: Tuesday, May 5th, 2026
– No teaching day (Substitution of classes)
★ (Week 12) Lecture #24: Wednesday, May 6th, 2026
– Iterative methods
– Jacobi iteration
– Gauss-Seidel iteration
– Relaxation
★ (Week 12) Practice #12: Wednesday, May 6th, 2026
– Iteration methods
★ (Week 13) Lecture #25: Tuesday, May 12th, 2026
– Structural review of all material needed for the exam
★ (Week 13) Lecture #26: Wednesday, May 13th, 2026
– Rector’s day
★ (Week 13) Practice #13: Wednesday, May 13th, 2026
– Rector’s day
★ (Week 14) Lecture #27: Tuesday, May 19th, 2026
– Free class to solve questions and doubts for the final exam
★ (Week 14) Lecture #28: Wednesday, May 20th, 2026
– Early bird exam
★ (Week 14) Practice #14: Wednesday, May 20th, 2026 (2 hours)
– Early bird exam (2 hours)
Please note that some exercises are intended for a more advanced part of the course. Try to focus on the ones we have already covered in class. The exercises are separated by topic.
Midterm: Tuesday, April 21st, 2026 (50 minutes to 80 minutes)
The midterm will consist of 4 problems that should verify the knowledge of basics on ODE theory.
50 minutes should be enough to complete the test.
Type of questions:
★ Solving an ODE using separation
★ Solving a homogeneous linear ODE
★ Guessing the form of solution for a linear ODE
★ Solving a system of homogeneous linear ODEs
HERE one can find the solutions of the Midterm
Final 1 (date)
Final 2 (date)
Final 3 (date)
Final 4 (date)
Final 5 (date)
Midterm – Tuesday, April 21st at 14.30 (Dejvice)
Final exam #1 (Early bird exam) – Wednesday, May 20th at 9.15 a.m. (Karlovo Náměstí)
Final exam #2
Final exam #3
Final exam #4
Final exam #5