Differential Equations for Engineers

The Hong Kong University of Science and Technology via Coursera

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Provider

Coursera
Pricing

Paid Course
Languages

English
Certificate

Paid Certificate Available
Duration & workload

1 day 2 hours 47 minutes
Sessions

On-Demand
Subtitles

Arabic, French, Portuguese, Italian, German, Russian, English, Spanish, Thai, Indonesian, Kazakh, Hindi, Swedish, Korean, Greek, Chinese, Ukrainian, Japanese, Polish, Dutch, Turkish, Hungarian, Bengali, Pashto, Urdu, Azerbaijani, Farsi

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Mathematics for Engineers

5.0

Overview

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This course is all about differential equations and covers both theory and applications. In the first five weeks, students will learn about ordinary differential equations, while the sixth week is an introduction to partial differential equations. The course includes 56 concise lecture videos, with a few problems to solve after each lecture. After each major topic, there is a short practice quiz. At the end of each week, there is an assessed quiz. Solutions to the problems and practice quizzes can be found in the instructor-provided lecture notes. Download the lecture notes from the link https://www.math.hkust.edu.hk/~machas/differential-equations-for-engineers.pdf Watch the promotional video from the link https://youtu.be/eSty7oo09ZI

Syllabus

First-Order Differential Equations

A differential equation is an equation for a function with one or more of its derivatives. We introduce different types of differential equations and how to classify them. We then discuss the Euler method for numerically solving a first-order ordinary differential equation (ODE). We learn analytical methods for solving separable and linear first-order ODEs, with an explanation of the theory followed by illustrative solutions of some simple ODEs. Finally, we explore three real-world examples of first-order ODEs: compound interest, the terminal velocity of a falling mass, and the resistor-capacitor electrical circuit.

Homogeneous Linear Differential Equations

We generalize the Euler numerical method to a second-order ODE. We then develop two theoretical concepts used for linear equations: the principle of superposition and the Wronskian. Using these concepts, we can find analytical solutions to a homogeneous second-order ODE with constant coefficients. We make use of an exponential ansatz and transform the constant-coefficient ODE to a second-order polynomial equation called the characteristic equation of the ODE. The characteristic equation may have real or complex roots and we learn solution methods for the different cases.

Inhomogeneous Linear Differential Equations

We now add an inhomogeneous term to the constant-coefficient ODE. The inhomogeneous term may be an exponential, a sine or cosine, or a polynomial. We also study the phenomena of resonance, when the forcing frequency is equal to the natural frequency of the oscillator. Finally, we learn about three important applications: the RLC electrical circuit, a mass on a spring, and the pendulum.

The Laplace Transform and Series Solution Methods

We present two new analytical solution methods for solving linear ODEs. The first is the Laplace transform method, which is used to solve the constant-coefficient ODE with a discontinuous or impulsive inhomogeneous term. The Laplace transform is a good vehicle in general for introducing sophisticated integral transform techniques within an easily understandable context. We also introduce the solution of a linear ODE by a series solution. Although we do not go deeply into it here, an introduction to this technique may be useful to students who encounter it again in more advanced courses.

Systems of Differential Equations

We learn how to solve a coupled system of homogeneous first-order differential equations with constant coefficients. This system of ODEs can be written in matrix form, and we learn how to convert these equations into a standard matrix algebra eigenvalue problem. The two-dimensional solutions are then visualized using phase portraits. We next learn about the important application of coupled harmonic oscillators and the calculation of normal modes. The normal modes are those motions for which the individual masses that make up the system oscillate with the same frequency. We then apply the theory to solve a system of two coupled harmonic oscillators, and use the normal modes to analyze the motion of the system.

Partial Differential Equations

To learn how to solve a partial differential equation (PDE), we first define a Fourier series. We then derive the one-dimensional diffusion equation, which is a PDE describing the diffusion of a dye in a pipe. We then proceed to solve this PDE using the method of separation of variables. This involves dividing the PDE into two ordinary differential equations (ODEs), which can then be solved using the standard techniques of solving ODEs. We then use the solutions of these two ODEs, and our definition of a Fourier series, to recover the solution of the original PDE.

Taught by

Jeffrey R. Chasnov

Reviews

4.9 rating, based on 381 Class Central reviews

4.9 rating at Coursera based on 2224 ratings

Start your review of Differential Equations for Engineers

Md. Tanvir Ahmed @tanvirece
2

Very easy to learn and fabulous technique of teaching for Differential Equations. I hope the instructor "Jeff Chasnov" will start the courses on Complex Variables, Co-ordinate Geometry, Probability and Statistics. It will be great pleasure for me if he will start these courses.
Soumya Acharya @strmclk
1

Great course, great overall coverage of topics, application-based examples aplenty. The instructor is really great at what he teaches. Worth the time and effort, especially if you are looking to simply learn/refresh your knowledge about DEs. Very satisfied overall with the learning; and there are other courses in an extension of this one that will be useful too (PDEs and numerical methods (the instructor is an author of a book on the latter that I've extensively used), for example.

As a pre-final year undergrad, I found it basic yet rigorous and ended up happily learning quite a few tricks I didn't initially set out to as part of my goals.
Anonymous

The Differential Equations for Engineers course by The Hong Kong University of Science and Technology on Coursera is very well structured and easy to follow. The lectures explain difficult concepts in a clear and simple way with practical engineering examples. The instructor, Jeffrey R. Chasnov, teaches with excellent clarity and provides helpful notes and exercises. Overall, it is a highly recommended course for students who want a strong foundation in differential equations.
Anonymous

This course provided a challenging yet rewarding deep dive into differential equations and Fourier methods. The content was well-structured, moving from fundamental ODE systems to PDEs with clear connections between theory and application. The assig…

This course provided a challenging yet rewarding deep dive into differential equations and Fourier methods. The content was well-structured, moving from fundamental ODE systems to PDEs with clear connections between theory and application. The assignments were rigorous, testing both conceptual understanding and problem-solving skills—especially in interpreting phase portraits and deriving series solutions. While some problem statements had slight ambiguities, working through them strengthened my analytical thinking. The mix of theoretical proofs and applied physics problems (like mass-spring systems and diffusion) made the material engaging and relevant. A highly recommended course for anyone serious about mastering mathematical methods for engineering and physics.
Ерасыл Рашбаев

This course provides a clear and structured introduction to differential equations, designed with engineers and applied scientists in mind. The professor explains the concepts step by step, balancing mathematical rigor with practical applications. T…

This course provides a clear and structured introduction to differential equations, designed with engineers and applied scientists in mind. The professor explains the concepts step by step, balancing mathematical rigor with practical applications. Topics range from first-order and second-order differential equations to systems, Fourier series, and partial differential equations, with examples tied to physics and engineering.
Strengths:
The explanations are concise and accessible, even for students without a very strong math background.
Video lectures are well-paced and supported by derivations, visualizations, and problem-solving demonstrations.
The quizzes and assignments test both conceptual understanding and practical problem-solving skills.
Applications to engineering contexts (heat diffusion, vibrations, circuits, etc.) make the theory more engaging.
Things to Note:
The course moves quickly, so reviewing calculus and linear algebra beforehand is very helpful.
Some problem sets can be challenging, but that makes the learning more rewarding.
Overall Impression:
This is an excellent course for anyone studying or working in engineering who wants a solid foundation in differential equations. It connects theory with application and prepares learners for more advanced studies in applied mathematics, control systems, and physics.
Rashbayev Yerassyl

This course provides a clear and structured introduction to differential equations, designed with engineers and applied scientists in mind. The professor explains the concepts step by step, balancing mathematical rigor with practical applications. T…

This course provides a clear and structured introduction to differential equations, designed with engineers and applied scientists in mind. The professor explains the concepts step by step, balancing mathematical rigor with practical applications. Topics range from first-order and second-order differential equations to systems, Fourier series, and partial differential equations, with examples tied to physics and engineering.
Strengths:
The explanations are concise and accessible, even for students without a very strong math background.
Video lectures are well-paced and supported by derivations, visualizations, and problem-solving demonstrations.
The quizzes and assignments test both conceptual understanding and practical problem-solving skills.
Applications to engineering contexts (heat diffusion, vibrations, circuits, etc.) make the theory more engaging.
Things to Note:
The course moves quickly, so reviewing calculus and linear algebra beforehand is very helpful.
Some problem sets can be challenging, but that makes the learning more rewarding.
Overall Impression:
This is an excellent course for anyone studying or working in engineering who wants a solid foundation in differential equations. It connects theory with application and prepares learners for more advanced studies in applied mathematics, control systems, and physics.
Anonymous

I really like it. it was well done and the professor explains everything perfectly. The quizzes are helpful and even the lecture notes. I would definitely recommend this course and this professor.
Anonymous

This course has allowed me to learn about the different applications and solution methods of ordinary differential equations (ODEs) and, especially, partial differential equations. For me, it has been a very enriching experience, both academically and personally. I sincerely thank the professor for his passion, clarity, and commitment to teaching, as well as the University and Coursera for providing this valuable learning opportunity.
Anonymous

The Differential Equations for Engineers course provides a clear and practical introduction to solving differential equations with engineering applications. It covers first- and second-order equations, Laplace transforms, and systems of ODEs with real-world examples that make abstract concepts easier to understand. The instructor explains complex topics step by step and connects math to physics and engineering problems. Practice exercises and quizzes reinforce learning effectively. Overall, it’s a well-structured and engaging course that strengthens both theoretical understanding and problem-solving skills — highly recommended for engineering students or anyone wanting to build strong fundamentals in differential equations.
Jan Willem

Excellent course by an Excellent teacher. Although I' more interested in maths, there are no equivalents for these subjects from a pure mathematical perspective on Coursera. But this (these) course(s) are great and provide a lot of insights, skills and a solid base to learn more.
Anonymous

This course is very well-structured and clear. The instructor explains the concepts step by step and connects theory with real engineering applications, which makes the subject much easier to understand. The examples and practice problems are very helpful for building confidence. I especially liked how the course shows the importance of differential equations in solving real-world problems, not just as abstract math. Overall, it is a high-quality course that I would strongly recommend to any engineering student or learner who wants to strengthen their foundation.
Anonymous

“From theory to applications, through clear explanations and challenging quizzes.”

I really enjoyed the course.
Anonymous

This course on Differential Equations for Engineers by HKUST via Coursera is a fantastic resource! It covers key concepts with clear examples and practical applications. The lectures are engaging, and the exercises reinforce learning effectively. Highly recommended for engineering students seeking a solid foundation.
Anonymous

This course not only provided me with basic knowledge of differential equations, but also introduced me to their practical application and gave my knowledge a boost. I hope this will help me in my future studies as an engineer.
Anonymous

This course provides the student with the powerful knowledge of differential equations for engineers. It is pretty useful for the basic course of differential equations and is very helpful for more detailed analysis of this subject
JUAN DE JESUS GOMEZ LOPEZ

Un curso en donde los temas son bastante bien explicados, claros y fácil de entender
Es una muy buena opción si quieres reforzar o aprender acerca de como resolver ecuaciones diferenciales
Anonymous

The course explains differential equations very clearly, starting from basic concepts to real engineering applications. The instructor uses intuitive explanations, examples, and visualizations that make abstract topics much easier to understand. The balance between theory and practical problem-solving is excellent. The quizzes and assignments reinforce learning well, and the pace is manageable even for beginners. Highly recommend for engineering students or anyone wanting a strong foundation in differential equations.
Anonymous

Very interesting the way the teacher it's just not giving any unknown formulas that come from a physics analysis and tackle the, but he reallym directly makes you learn in a very simple way how to truly understand mathemathics while you're learning differential equations in the road, always viewed from a real physics perspective. Covered the most important subjects gave enough material to practice. Excellent and congratulations.
Anavheoba Abraham Ogenakohgie
2

Professor Jeff chaznov really did a good job in taking this course (Differential equation for engineers) At first when he started he went back to basic calculus and was nurturing us like babies(for the first three weeks of the course) and I apprecia…

Professor Jeff chaznov really did a good job in taking this course (Differential equation for engineers)
At first when he started he went back to basic calculus and was nurturing us like babies(for the first three weeks of the course) and I appreciate that a lot. It was easy at first in the first three weeks but it became complex due to the introduction of Laplace transform,Heaviside's step function,Airy's expansion it was really fun cause I learnt a lot and I would love to know more.
This course was really challenging it made me think outside the Box,and I appreciate prof for tutoring this course like this.
I can boast now that there is nothing in calculus that I don't have knowledge of,because of the way this course was treated. I'm still in college but I believe this course will prove very useful in the future.
Thank you very much Jeff chaznov
It was worth it cause u changed lives
Anonymous

if you already know the prerequisite math, this course will be a breeze for you. it was challenging at times but having the intuition to break down problems makes it easier to understand the work you're doing.