Differential Equations for Engineers from Coursera

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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.

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Explores differential equations, a fundamental concept in engineering, science, and technology

Taught by Jeffrey R. Chasnov, a respected professor with research interests in differential equations, partial differential equations, and applications of mathematics

Develops a strong foundation in the theory and application of differential equations, making it suitable for both beginners and intermediate learners

Covers a wide range of topics, including first-order differential equations, linear differential equations, Laplace transforms, and partial differential equations

Provides practice problems, quizzes, and lecture notes to enhance understanding and reinforce concepts

Requires a strong background in calculus and linear algebra

Reviews summary

Clear & concise differential equations

learners say this course on differential equations is largely positive, with engaging assignments, well designed lectures, and an esteemed instructor. This course is highly-recommended for engineers and students in related fields. It will be especially helpful for learners with a foundation in integral calculus.

learners found the course to be practical and applicable to real-world problems in engineering and other fields. The coverage of a wide range of topics was also seen as a strength of the course.

"Great course to learn about Differential Equations for Engineers."

"I am with the university of Benin,I just completed this course but I have not received any certificatePlease I really need this certificate as soon as possibleThanks"

"This course is very helpful for civil engineers. Since I am currently a CE student, this will be my guide to practice DE more. Special thanks also to the instructor!"

"This course was very good and it helped me to know and learn a lot about differential equations. The instructor as well as the class notes was great . Thank you"

learners found the course materials to be high-quality and supportive of the learning process. The lecture notes were praised for their clarity and completeness.

"This course is very very useful for my teaching and I improve my self. Differential Equations is one of my favorite topic.I can do another course also."

"Engaging and high-quality teaching. Very well structured and a good mix of short lectures and corresponding exercises. Thanks for putting this together!"

"Hello, thank you for your efforts in the course. It was very useful and beautiful. I hope you will always be healthy and happy. Mohammad Pakzad from Iran."

"This course is very helpful for my teaching and I improve my self. Differential Equations is one of my favorite topic.I can do another course also."

Learners found this course to be well-paced and organized, with clear and concise explanations of key concepts. The assignments were described as challenging but fair, and the instructor was praised for his enthusiasm and availability to answer questions.

"The course went beyond my expectations! It was really worthwhile taking some time to participate."

"This course is very useful and each in engineering students. Thank you very much for giving this course."

"It was a very good introduction. I only missed the theory differential equations with non-constant coefficients."

"Professor Chasnov is outstanding. He explains concepts well and has a lot of passion for the topic. "

learners felt that the course was well-structured and well-paced, with a logical progression of topics. The inclusion of practice quizzes was also seen as beneficial, as it allowed learners to test their understanding of the material.

"I have received quite clear explanation and I can do all the practice quiz and reading problems well. Thanks, professor."

"Great course! Lectures are engaging, lecture notes are very useful! Quiz questions are challenging but they are helpful!"

"This is a really recommended step-by-step learning of differential equation. The hard material are divided into pieces which are really easy to follow through."

"The way of teaching and explanation is excellent. By taking this course I really enhanced my teaching skills. I express my sincere thanks to Prof. Jeffery R Chasnov from bottom of my heart."

Professor Jeffrey Chasnov was lauded for his engaging and knowledgeable teaching style. Learners appreciated his ability to break down complex topics into manageable chunks and his willingness to answer questions. His passion for the subject matter was also evident.

"This course is well designed, with detailed explanations and well-structured examples. Course difficulty is fair, definitely recommended for those who want to learn differential equations with basic college math knowledge."

"I took this course as a refresher before I start my PHD and it definitely gave me what I wanted. It's simple enough to work well for beginners and comprehensive enough to give you all you need to start working more difficult topics."

"Solving the diffusion equation by Fourier series is very illuminating."

"Although personally speaking this course is the most difficult one in this series, the course arrangement is systematic and reasonable. Students can easily follow the course."

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Differential Equations for Engineers with these activities:

Organize and Review Course Materials

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Stay organized and reinforce your learning by reviewing and compiling course materials.

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Collect and organize all lecture notes, assignments, and other course materials.
Review the materials regularly to refresh your understanding.
Identify areas where you need further clarification and seek additional resources.

Differential Equations Review

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Review the fundamentals of differential equations to strengthen your foundation for this course.

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Review your notes or textbook from previous courses on differential equations.
Solve practice problems to test your understanding.

Follow YouTube Tutorials on Differential Equations

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Reinforce your understanding of concepts by watching and following along with video tutorials.

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Search for YouTube videos on specific topics you want to review.
Take notes and pause the videos to solve problems yourself.
Review the videos multiple times to reinforce your understanding.

Five other activities

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Solve Differential Equation Practice Problems

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Improve your problem-solving skills and strengthen your understanding of different techniques.

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Find practice problems from textbooks or online resources.
Allocate dedicated time to solving the problems.
Check your solutions against provided answers or consult with the instructor for feedback.

Create a Summary of Differential Equation Concepts

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Solidify your understanding by creating a comprehensive summary of key concepts.

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Review your notes and identify the most important concepts.
Organize the concepts into a logical structure.
Write a clear and concise summary using your own words.

Join a Study Group for Differential Equations

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Collaborate with peers to discuss concepts, solve problems, and enhance your understanding.

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Find or create a study group with classmates.
Establish a regular meeting schedule.
Take turns presenting concepts and leading discussions.

Participate in Differential Equation Competitions or Hackathons

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Challenge yourself and test your skills by participating in competitions or hackathons.

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Research and find differential equation competitions or hackathons.
Form a team or participate individually.
Practice solving differential equations and develop strategies.

Mentor a Junior or High School Student in Differential Equations

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Enhance your understanding by teaching and guiding others, while making a positive impact.

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Identify a student who needs help with differential equations.
Establish a regular tutoring schedule.
Prepare lesson plans and materials to support the student's learning.

Career center

Learners who complete Differential Equations for Engineers will develop knowledge and skills that may be useful to these careers:

Electrical Engineer

Electrical Engineers design, develop, test, and supervise the installation of electrical systems. Differential equations are used in many aspects of electrical engineering, such as circuit analysis, power systems, and control systems. This course provides a strong foundation in differential equations, which can help an Electrical Engineer build the skills needed to analyze and design electrical systems effectively.

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Control Systems Engineer

Control Systems Engineers design systems to manage and control a wide variety of processes, from manufacturing to energy production. Engineers in this role may use differential equations to model and analyze control systems, identify potential problems, and develop solutions to improve system performance. This course provides a solid foundation in differential equations, which can help a Control Systems Engineer build the skills necessary to design and implement effective control systems.

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Physicist

Physicists study the fundamental laws of nature. Differential equations are used in many areas of physics, such as mechanics, electromagnetism, and quantum mechanics. This course provides a strong foundation in differential equations, which can help a Physicist build the skills needed to understand and describe the physical world.

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Mechanical Engineer

Mechanical Engineers design, develop, and test mechanical systems, such as engines, robots, and medical devices. Differential equations are used in many aspects of mechanical engineering, such as fluid mechanics, heat transfer, and structural analysis. This course provides a strong foundation in differential equations, which can help a Mechanical Engineer build the skills needed to analyze and design mechanical systems effectively.

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Systems Engineer

Systems Engineers design, develop, and maintain complex systems, such as computer networks, manufacturing systems, and transportation systems. Differential equations are used in many aspects of systems engineering, such as system modeling, simulation, and control. This course provides a strong foundation in differential equations, which can help a Systems Engineer build the skills needed to analyze and design complex systems effectively.

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Petroleum Engineer

Petroleum Engineers design and develop methods for extracting and producing oil and gas. Differential equations are used in many aspects of petroleum engineering, such as reservoir simulation, wellbore analysis, and production optimization. This course provides a strong foundation in differential equations, which can help a Petroleum Engineer build the skills needed to analyze and design petroleum systems effectively.

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Data Scientist

Data Scientists use their knowledge of mathematics and statistics to extract insights from data and solve business problems. Differential equations are used in a variety of data science applications, such as time series analysis, forecasting, and modeling complex systems. This course introduces the fundamental concepts of differential equations, which can help a Data Scientist build the skills needed to succeed in this field.

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Technical Writer

Technical Writers create and edit technical documentation, such as manuals, reports, and articles. Differential equations are often used in technical documentation. This course provides a solid foundation in differential equations, which can help a Technical Writer understand and communicate technical concepts effectively.

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Operations Research Analyst

Operations Research Analysts use their knowledge of mathematics and statistics to solve complex business problems. Differential equations are used in a variety of operations research applications, such as scheduling, inventory management, and supply chain optimization. This course provides a solid foundation in differential equations, which can help an Operations Research Analyst build the skills needed to succeed in this field.

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Statistician

Statisticians collect, analyze, and interpret data to help businesses and governments make informed decisions. Differential equations are used in a variety of statistical applications, such as survival analysis, queueing theory, and image processing. This course provides a solid foundation in differential equations, which can help a Statistician build the skills needed to analyze and interpret data effectively.

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Software Engineer

Software Engineers design, develop, and maintain computer software. Differential equations are used in a variety of software applications, such as computer graphics, artificial intelligence, and robotics. This course provides a solid foundation in differential equations, which can help a Software Engineer build the skills needed to develop software that is efficient and effective.

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Financial Analyst

Financial Analysts use their knowledge of mathematics and economics to analyze financial data and make investment recommendations. Differential equations are used in a variety of financial applications, such as stock pricing, risk management, and portfolio optimization. This course provides a solid foundation in differential equations, which can help a Financial Analyst build the skills needed to succeed in this field.

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Web Developer

Web Developers design and develop websites. Differential equations are used in some web development applications, such as animation and interactive simulations. This course provides a foundation in differential equations, which can help a Web Developer build the skills needed to develop more engaging and dynamic websites.

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Teacher

Teachers educate students at all levels, from kindergarten through college. Differential equations are taught in many high school and college math classes. This course provides a strong foundation in differential equations, which can help a Teacher prepare to teach this subject to students.

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Writer

Writers create and edit written content, such as books, articles, and speeches. Differential equations are sometimes used in creative writing, for example, to describe the motion of a character in a story. This course provides a foundation in differential equations, which can help a Writer understand and use this tool effectively in their writing.

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