Introduction to Ordinary Differential Equations
2025-fall, Università della Svizzera italiana, Faculty of Informatics, 2025
This course introduces the fundamental concepts and techniques of ordinary differential equations (ODEs), with a focus on applications in artificial intelligence and computational science. We connect modeling and analysis (existence, stability, qualitative behavior) with computation (numerical initial-value solvers, stiffness) and modern ODE-based learning (Neural ODEs, sparse discovery, and multiscale/network dynamics).
For up-to-date information on schedule and lecture room, visit the official USI course page.
Schedule (14 weeks)
- Format: 1 lecture per week on Fridays. After every two lectures there is an Exercises + Quiz session (first half TA Q&A, second half quiz). After Quiz 4 there is one last lecture and a final Exercises session before the exam.
| Week | Date | Topic | Materials | Notes |
|---|---|---|---|---|
| 1 | 2025-09-19 | Lecture 1: Introduction and First-Order Equations | Notes | |
| 2 | 2025-09-26 | Lecture 2: Systems of First-Order Equations | Notes | |
| 3 | 2025-10-03 | Exercises + Quiz 1 | TA Q&A + Quiz | |
| 4 | 2025-10-10 | Lecture 3: Linear Systems and Matrix Methods | Notes | |
| 5 | 2025-10-17 | Lecture 4: Eigenvalue Methods and Diagonalization | Notes | |
| 6 | 2025-10-24 | Exercises + Quiz 2 | TA Q&A + Quiz | |
| 7 | 2025-10-31 | Lecture 5: Nonlinear Dynamics and Phase Plane Analysis | Notes | |
| 8 | 2025-11-07 | Lecture 6: Stability Theory and Lyapunov Methods | Notes | |
| 9 | 2025-11-14 | Exercises + Quiz 3 | TA Q&A + Quiz | |
| 10 | 2025-11-21 | Lecture 7: Numerical Methods for Differential Equations | Notes | |
| 11 | 2025-11-28 | Lecture 8: Applications in Science and Engineering | Notes | |
| 12 | 2025-12-05 | Exercises + Quiz 4 | TA Q&A + Quiz | |
| 13 | 2025-12-12 | Lecture 9: Advanced Topics and Current Research | Notes | |
| 14 | 2025-12-19 | Final Exercises & Q&A (exam preparation) | No quiz |
Slight adjustments may occur; any changes will be announced here and in class.
Course Outline (Lectures 1–9)
- Lecture 1 — Introduction and First-Order Equations: Basic concepts, classification of ODEs, direction fields, isoclines, separable equations, and linear first-order equations with integrating factors.
- Lecture 2 — Systems of First-Order Equations: Reduction of higher-order equations, phase space, trajectories, nullclines, and linearization for analyzing equilibria in systems like predator-prey models.
- Lecture 3 — Linear Systems and Matrix Methods: The matrix exponential, eigenvalue analysis for classifying 2D systems (nodes, saddles, spirals), fundamental matrices, and nonhomogeneous systems.
- Lecture 4 — Eigenvalue Methods and Diagonalization: Solving systems via diagonalization and modal coordinates, handling complex and repeated eigenvalues, and applications to mechanical systems.
- Lecture 5 — Nonlinear Dynamics and Phase Plane Analysis: In-depth analysis of nonlinear systems, limit cycles, Poincaré-Bendixson theorem, and local bifurcations (saddle-node, transcritical, pitchfork, Hopf).
- Lecture 6 — Stability Theory and Lyapunov Methods: Rigorous definitions of stability, Lyapunov’s direct method, LaSalle’s invariance principle, estimating basins of attraction, and stability of periodic orbits.
- Lecture 7 — Numerical Methods for Differential Equations: Euler’s method, Runge-Kutta methods, multi-step methods, local vs. global error, stability regions, stiffness, adaptive step-size control, and geometric integration.
- Lecture 8 — Applications in Science and Engineering: Broad survey of ODE modeling in mechanics, population dynamics (logistic growth, SIR models), biochemical networks, and coupled oscillator systems.
- Lecture 9 — Advanced Topics and Current Research: Modern developments including Neural ODEs with adjoint sensitivity, sparse discovery of dynamics (SINDy), consensus on graphs, multiscale methods, and continuous normalizing flows.