TÜRKÇE
ENGLISH
Information
| Unit | FACULTY OF SCIENCE AND LETTERS |
| ARTIFICIAL INTELLIGENCE AND MACHINE LEARNING PR. (ENGLISH) | |
| Code | YZZ201 |
| Name | Differential Equations |
| Term | 2026-2027 Academic Year |
| Semester | 3. Semester |
| Duration (T+A) | 3-0 (T-A) (17 Week) |
| ECTS | 4 ECTS |
| National Credit | 3 National Credit |
| Teaching Language | İngilizce |
| Level | Lisans Dersi |
| Type | Normal |
| Label | C Compulsory |
| Mode of study | Yüz Yüze Öğretim |
| Catalog Information Coordinator | Prof. Dr. YUSUF ALPER KAPLAN |
| Course Instructor |
Dr. Öğr. Üyesi Kasım ZOR
(Güz)
(A Group)
(Ins. in Charge)
|
Course Goal / Objective
This course aims to teach students differential equations (DEs) and their applications in artificial intelligence and machine learning along with several engineering disciplines.
Course Content
Introduction to DEs, First-Order Ordinary Differential Equations (ODEs), Second-Order Linear ODEs, Higher-Order Linear ODEs, The Laplace Transform, Systems of First-Order ODEs, Partial DEs and Series Solutions, Applications in AI&ML: Neural ODEs and Gradient Flow.
Course Precondition
There are no formal prerequisites for the course. (Recommended foundational courses: YZZ101 Mathematics I, YZZ102 Mathematics II, and YZZ105 Linear Algebra).
Resources
David V. Kalbaugh, Differential Equations for Engineers: The Essentials, 1st Ed., CRC Press, 2018.
Notes
Joakim Sundnes, Solving Ordinary Differential Equations in Python, 1st Ed., Springer, 2024. Karline Soetaert, Jeff Cash, and Francesca Mazzia, Solving Differential Equations in R, 1st Ed., Springer 2012. Clemens Heitzinger, Algorithms with JULIA: Optimisation, Machine Learning, and Differential Equations Using the JULIA Language, 1st Ed., Springer, 2022. Christian Constanda, Differential Equations: A Primer for Scientists and Engineers, 2nd Ed., Springer, 2017. Dennis G. Zill, Advanced Engineering Mathematics, 6th Ed., Jones & Bartlett Learning, 2018. William E. Boyce, Richard C. Diprima, and Douglas B. Meade, Elementary Differential Equations and Boundary Value Problems, 11th Ed., Wiley, 2017. Brent J. Lewis, E. Nihan Onder, and Andrew A. Prudil, Advanced Mathematics for Engineering Students: The Essential Toolbox, 1st Ed., Elsevier, 2022. Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David Duvenaud, Neural Ordinary Differential Equations, Advances in Neural Information Processing Systems (NeurIPS), 2018. Christopher Rackauckas, SciML Book: Parallel Computing and Scientific Machine Learning, MIT (open access, online), 2020.
Course Learning Outcomes
| Order | Course Learning Outcomes |
|---|---|
| LO01 | Classify differential equations by order, linearity, and type (ordinary vs. partial), and identify the appropriate solution method for a given problem. |
| LO02 | Formulate and solve first-order ODEs analytically using separation of variables, integrating factors, and exact equation techniques. |
| LO03 | Solve second-order and higher-order linear ODEs using methods including undetermined coefficients, variation of parameters, and series solutions. |
| LO04 | Apply the Laplace transform to solve linear ODEs with discontinuous or impulsive forcing functions, including initial value problems. |
| LO05 | Formulate and solve systems of first-order ODEs, and analyse the stability and qualitative behaviour of their solutions. |
| LO06 | Derive and interpret basic partial differential equations and their series-based solution methods, including separation of variables. |
| LO07 | Implement numerical solvers for ODEs and systems of ODEs using at least one computational tool (Python, R, or Julia), and critically compare analytical and numerical solutions. |
| LO08 | Explain the mathematical connection between differential equations and gradient-based optimisation methods used in machine learning, including the interpretation of gradient descent as a continuous-time dynamical system. |
| LO09 | Describe the formulation of Neural ODEs and articulate their relevance to continuous-depth deep learning architectures. |
| LO10 | Evaluate the applicability of differential equation-based modelling to engineering and artificial intelligence problems, and select suitable analytical or computational approaches for novel problem settings. |
Relation with Program Learning Outcome
| Order | Type | Program Learning Outcomes | Level |
|---|---|---|---|
| PLO01 | Bilgi - Kuramsal, Olgusal | It provides a broad range of knowledge about fundamental Computer Science concepts, algorithms and data structures. | 3 |
| PLO02 | Bilgi - Kuramsal, Olgusal | Learns basic computer topics such as software development, programming languages, and database management. | 3 |
| PLO03 | Bilgi - Kuramsal, Olgusal | Understands advanced computing fields such as data science, artificial intelligence, and machine learning. | 3 |
| PLO04 | - | Learn about topics such as computer networks, cyber security, and database design. | |
| PLO05 | Beceriler - Bilişsel, Uygulamalı | Develops skills in designing, implementing and analyzing algorithms. | 4 |
| PLO06 | Beceriler - Bilişsel, Uygulamalı | Gains the ability to use different programming languages effectively | |
| PLO07 | Beceriler - Bilişsel, Uygulamalı | Learns data analysis, database management and big data processing skills. | |
| PLO08 | Beceriler - Bilişsel, Uygulamalı | Gains practical experience by working on software development projects. | |
| PLO09 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | Strengthens collaboration and communication skills within the team. | |
| PLO10 | Yetkinlikler - Alana Özgü Yetkinlik | It provides a mindset open to technological innovations. | |
| PLO11 | Yetkinlikler - Öğrenme Yetkinliği | Encourages continuous learning and self-improvement competence. | |
| PLO12 | Yetkinlikler - İletişim ve Sosyal Yetkinlik | Develops the ability to solve complex problems. | 4 |
Week Plan
| Week | Topic | Preparation | Methods |
|---|---|---|---|
| 1 | Course Introduction and Scope, Foundations of Differential Equations | Reading the lecture notes | Öğretim Yöntemleri: Anlatım, Soru-Cevap, Tartışma, Alıştırma ve Uygulama, Problem Çözme |
| 2 | First-Order Linear Ordinary Differential Equations | Reading the lecture notes | Öğretim Yöntemleri: Anlatım, Soru-Cevap, Tartışma, Alıştırma ve Uygulama, Problem Çözme |
| 3 | First-Order Nonlinear Ordinary Differential Equations | Reading the lecture notes | Öğretim Yöntemleri: Anlatım, Soru-Cevap, Tartışma, Alıştırma ve Uygulama, Problem Çözme |
| 4 | Existence and Uniqueness | Reading the lecture notes | Öğretim Yöntemleri: Anlatım, Soru-Cevap, Tartışma, Alıştırma ve Uygulama, Problem Çözme |
| 5 | Second-Order Linear Ordinary Differential Equations | Reading the lecture notes | Öğretim Yöntemleri: Anlatım, Soru-Cevap, Tartışma, Alıştırma ve Uygulama, Problem Çözme |
| 6 | Higher-Order Linear Ordinary Differential Equations | Reading the lecture notes | Öğretim Yöntemleri: Anlatım, Soru-Cevap, Tartışma, Alıştırma ve Uygulama, Problem Çözme |
| 7 | Higher-Order Linear Ordinary Differential Equations - 2 | Reading the lecture notes | Öğretim Yöntemleri: Anlatım, Soru-Cevap, Tartışma, Alıştırma ve Uygulama, Problem Çözme |
| 8 | Midterm Examination | Reviewing the previous topics | Ölçme Yöntemleri: Yazılı Sınav |
| 9 | Laplace Transforms | Reading the lecture notes | Öğretim Yöntemleri: Anlatım, Soru-Cevap, Tartışma, Alıştırma ve Uygulama, Problem Çözme |
| 10 | Systems of First-Order Ordinary Differential Equations | Reading the lecture notes | Öğretim Yöntemleri: Anlatım, Soru-Cevap, Tartışma, Alıştırma ve Uygulama, Problem Çözme |
| 11 | Systems of First-Order Ordinary Differential Equations - 2 | Reading the lecture notes | Öğretim Yöntemleri: Anlatım, Soru-Cevap, Tartışma, Alıştırma ve Uygulama, Problem Çözme |
| 12 | Partial Differential Equations and Series Solutions | Reading the lecture notes | Öğretim Yöntemleri: Anlatım, Soru-Cevap, Tartışma, Alıştırma ve Uygulama, Problem Çözme |
| 13 | Partial Differential Equations and Series Solutions - 2 | Reading the lecture notes | Öğretim Yöntemleri: Anlatım, Soru-Cevap, Tartışma, Alıştırma ve Uygulama, Problem Çözme |
| 14 | Applications in AI&ML: Neural ODEs and Gradient Flow | Reading the lecture notes | Öğretim Yöntemleri: Anlatım, Tartışma, Soru-Cevap, Alıştırma ve Uygulama, Problem Çözme |
| 15 | Applications of Differential Equations in AI&ML: Neural ODEs and Gradient Flow. - 2 | Reading the lecture notes | Ölçme Yöntemleri: Yazılı Sınav |
| 16 | Final Examination | Reviewing the previous topics | Ölçme Yöntemleri: Yazılı Sınav |
| 17 | Final Examination | Reviewing the previous topics | Ölçme Yöntemleri: Yazılı Sınav |
Student Workload - ECTS
| Works | Number | Time (Hour) | Workload (Hour) |
|---|---|---|---|
| Course Related Works | |||
| Class Time (Exam weeks are excluded) | 14 | 3 | 42 |
| Out of Class Study (Preliminary Work, Practice) | 14 | 3 | 42 |
| Assesment Related Works | |||
| Homeworks, Projects, Others | 12 | 1 | 12 |
| Mid-term Exams (Written, Oral, etc.) | 1 | 2 | 2 |
| Final Exam | 1 | 2 | 2 |
| Total Workload (Hour) | 100 | ||
| Total Workload / 25 (h) | 4,00 | ||
| ECTS | 4 ECTS | ||