Differential Equations for Engineers

Go to Course: https://www.coursera.org/learn/differential-equations-engineers

Introduction

# Course Review: Differential Equations for Engineers on Coursera Looking for a comprehensive and insightful introduction to the world of differential equations? **Differential Equations for Engineers**, offered on Coursera, stands out as a meticulously crafted course that combines both the theoretical foundations and practical applications of this critical area of mathematics. As an aspiring engineer or a seasoned professional seeking to refine your knowledge, this course has much to offer. ## Course Overview This course spans six weeks and delves deeply into the realm of differential equations, primarily catering to engineering students or professionals. The structure is methodical, beginning with ordinary differential equations (ODEs) and transitioning to partial differential equations (PDEs) in the final week. With a total of **56 concise lecture videos**, the course promises a rich learning experience without overwhelming the learner. One of the best features of the course is its integration of problem-solving exercises after each lecture, allowing you to apply and consolidate your understanding in real-time. The course includes practice quizzes after major topics and assessed quizzes at the end of each week, ensuring that you are keeping up with the material and grasping the core concepts. ## Course Syllabus Highlights ### 1. **First-Order Differential Equations** The course kicks off with an introduction to the fundamental concept of a differential equation. You will learn how to classify different types of equations, the Euler method for numerical solutions, and analytical methods for solving separable and linear first-order ODEs. Real-world applications such as compound interest and electrical circuits make the content relatable and practically valuable. ### 2. **Homogeneous Linear Differential Equations** Students are guided through the principles of superposition and Wronskian, leading to solutions of second-order ODEs with constant coefficients. This section is crucial as it lays a solid foundation for understanding more complex equations later on. ### 3. **Inhomogeneous Linear Differential Equations** This part dives into adding inhomogeneous terms to ODEs and explores important applications, including RLC circuits and pendulums. The exploration of resonance phenomena is particularly fascinating and demonstrates the power of differential equations in real-world systems. ### 4. **The Laplace Transform and Series Solution Methods** Students are introduced to the Laplace transform, an invaluable technique for tackling ODEs with impulsive inhomogeneous terms. The discussion on series solution methods provides a gateway into more sophisticated analytical techniques. ### 5. **Systems of Differential Equations** This week focuses on solving coupled systems of first-order ODEs, introducing matrix algebra and eigenvalue problems. Practical applications, such as normal modes in coupled harmonic oscillators, highlight the utility of these concepts in engineering. ### 6. **Partial Differential Equations** The final week addresses PDEs, beginning with Fourier series and moving on to the one-dimensional diffusion equation. The method of separation of variables offers a structured approach to tackling complex equations. ## Why You Should Take This Course 1. **Clear Structure and Pacing**: The course is thoughtfully organized, allowing you to build upon your knowledge week by week. 2. **Practical Applications**: Each theoretical concept is accompanied by real-world applications, which is essential for engineers who need to see the relevance of their studies. 3. **Interactive Learning**: With practice problems and quizzes after most lectures, the opportunity to actively engage with the material—and the immediate feedback—enhance learning outcomes. 4. **Expert Instruction**: The course is taught by knowledgeable instructors who make complex topics more approachable and understandable. 5. **Suitable for All Levels**: Whether you are a beginner or someone looking to refresh your knowledge, this course is adaptable to different levels of expertise. ## Conclusion **Differential Equations for Engineers** on Coursera is not just a course; it is a pathway to mastering an essential mathematical tool of engineering. By offering a balanced mix of theory and practical problems, this course prepares students for real-world applications, ensuring they leave with a thorough understanding of differential equations. I highly recommend this course for anyone looking to deepen their mathematical skills, particularly in engineering contexts. Enroll today, and embark on your journey to becoming proficient in differential equations!

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.

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.

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.

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.

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.

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.