Offered By

The Hong Kong University of Science and Technology

About this Course

This is a course is about differential equations, and covers material that all engineers should know. We will learn how to solve first-order equations, and how to solve second-order equations with constant coefficients and also look at some fundamental engineering applications. We will learn about the Laplace transform and series solution methods. Finally, we will learn about systems of linear differential equations, including the very important normal modes problem, and how to solve a partial differential equation using separation of variables. This solution method requires first learning about Fourier series.
After each video, there are problems to solve and I have tried to choose problems that exemplify the main idea of the lecture. I try to give enough problems for students to solidify their understanding of the material, but not so many that students feel overwhelmed. I do encourage students to attempt the given problems, but if they get stuck, full solutions can be found in the lecture notes for the course.
Lecture notes may be downloaded at
http://www.math.ust.hk/~machas/differential-equations-for-engineers.pdf

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Knowledge of single variable calculus.

Suggested: 5 hours per week...

Subtitles: English

Start instantly and learn at your own schedule.

Reset deadlines in accordance to your schedule.

Knowledge of single variable calculus.

Suggested: 5 hours per week...

Subtitles: English

Week

1Welcome to the first module! We begin by introducing differential equations and classifying them. We then explain the Euler method for numerically solving a first-order ode. Next, we explain the analytical solution methods for separable and linear first-order odes. An explanation of the theory is followed by illustrative solutions of some simple odes. Finally, we present three real-world examples of first-order odes and their solution: compound interest, terminal velocity of a falling mass, and the resistor-capacitor electrical circuit. ...

12 videos (Total 97 min), 11 readings, 6 quizzes

Course Overview2m

Introduction to Differential Equations9m

Week 1 Introduction47s

Euler Method9m

Separable First-order Equations8m

Separable First-order Equation: Example6m

Linear First-order Equations13m

Linear First-order Equation: Example5m

Application: Compound Interest13m

Application: Terminal Velocity11m

Application: RC Circuit11m

Welcome and Course Information2m

Get to Know Your Classmates10m

Practice: Runge-Kutta Methods10m

Practice: Separable First-order Equations10m

Practice: Separable First-order Equation Examples10m

Practice: Linear First-order Equations5m

A Change of Variables Can Convert a Nonlinear Equation to a Linear equation10m

Practice: Linear First-order Equation: Examples10m

Practice: Compound Interest10m

Practice: Terminal Velocity10m

Practice: RC Circuit10m

Diagnostic Quiz15m

Classify Differential Equations10m

Separable First-order ODEs15m

Linear First-order ODEs15m

Applications20m

Week Ones

Week

2We begin by generalising the Euler numerical method to a second-order equation. We then develop two theoretical concepts used for linear equations: the principle of superposition, and the Wronskian. Armed with these concepts, we can find analytical solutions to a homogeneous second-order ode with constant coefficients. We make use of an exponential ansatz, and convert the 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 discuss the solutions for these different cases. We then consider the inhomogeneous ode, and the phenomena of resonance, where the forcing frequency is equal to the natural frequency of the oscillator. Finally, some interesting and important applications are discussed....

22 videos (Total 218 min), 20 readings, 3 quizzes

Euler Method for Higher-order ODEs9m

The Principle of Superposition6m

The Wronskian8m

Homogeneous Second-order ODE with Constant Coefficients9m

Case 1: Distinct Real Roots7m

Case 2: Complex-Conjugate Roots (Part A)7m

Case 2: Complex-Conjugate Roots (Part B)8m

Case 3: Repeated Roots (Part A)12m

Case 3: Repeated Roots (Part B)4m

Inhomogeneous Second-order ODE9m

Inhomogeneous Term: Exponential Function11m

Inhomogeneous Term: Sine or Cosine (Part A)9m

Inhomogeneous Term: Sine or Cosine (Part B)8m

Inhomogeneous Term: Polynomials7m

Resonance13m

RLC Circuit11m

Mass on a Spring9m

Pendulum12m

Damped Resonance14m

Complex Numbers17m

Nondimensionalization17m

Practice: Second-order Equation as System of First-order Equations10m

Practice: Second-order Runge-Kutta Method10m

Practice: Linear Superposition for Inhomogeneous ODEs10m

Practice: Wronskian of Exponential Function10m

Do You Know Complex Numbers?

Practice: Roots of the Characteristic Equation10m

Practice: Distinct Real Roots10m

Practice: Hyperbolic Sine and Cosine Functions10m

Practice: Complex-Conjugate Roots10m

Practice: Sine and Cosine Functions10m

Practice: Repeated Roots10m

Practice: Multiple Inhomogeneous Terms10m

Practice: Exponential Inhomogeneous Term10m

Practice: Sine or Cosine Inhomogeneous Term10m

Practice: Polynomial Inhomogeneous Term10m

When the Inhomogeneous Term is a Solution of the Homogeneous Equation10m

Do You Know Dimensional Analysis?

Another Nondimensionalization of the RLC Circuit Equation10m

Another Nondimensionalization of the Mass on a Spring Equation10m

Find the Amplitude of Oscillation10m

Homogeneous Equations20m

Inhomogeneous Equations20m

Week Twos

Week

3We 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 series solution. Although we do not go deeply here, an introduction to this technique may be useful to students that encounter it again in more advanced courses. ...

11 videos (Total 123 min), 10 readings, 4 quizzes

Definition of the Laplace Transform13m

Laplace Transform of a Constant Coefficient ODE11m

Solution of an Initial Value Problem13m

The Heaviside Step Function10m

The Dirac Delta Function12m

Solution of a Discontinuous Inhomogeneous Term13m

Solution of an Impulsive Inhomogeneous Term7m

The Series Solution Method17m

Series Solution of the Airy's Equation (Part A)14m

Series Solution of the Airy's Equation (Part B)7m

Practice: The Laplace Transform of Sine10m

Practice: Laplace Transform of an ODE10m

Practice: Solution of an Initial Value Problem10m

Practice: Heaviside Step Function10m

Practice: The Dirac Delta Function15m

Practice: Discontinuous Inhomogeneous Term20m

Practice: Impulsive Inhomogeneous Term10m

Practice: Series Solution Method10m

Practice: Series Solution of a Nonconstant Coefficient ODE1m

Practice: Solution of the Airy's Equation10m

The Laplace Transform Method30m

Discontinuous and Impulsive Inhomogeneous Terms20m

Series Solutions20m

Week Threes

Week

4We solve a coupled system of homogeneous linear first-order differential equations with constant coefficients. This system of odes can be written in matrix form, and we explain how to convert these equations into a standard matrix algebra eigenvalue problem. We then discuss 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. Next, to prepare for a discussion of partial differential equations, we define the Fourier series of a function. Then we derive the well-known one-dimensional diffusion equation, which is a partial differential equation for the time-evolution of the concentration of a dye over one spatial dimension. We proceed to solve this equation for a dye diffusing length-wise within a finite pipe.
...

19 videos (Total 177 min), 17 readings, 6 quizzes

Systems of Homogeneous Linear First-order ODEs8m

Distinct Real Eigenvalues9m

Complex-Conjugate Eigenvalues12m

Coupled Oscillators9m

Normal Modes (Eigenvalues)10m

Normal Modes (Eigenvectors)9m

Fourier Series12m

Fourier Sine and Cosine Series5m

Fourier Series: Example11m

The Diffusion Equation9m

Solution of the Diffusion Equation: Separation of Variables11m

Solution of the Diffusion Equation: Eigenvalues10m

Solution of the Diffusion Equation: Fourier Series9m

Diffusion Equation: Example10m

Matrices and Determinants13m

Eigenvalues and Eigenvectors10m

Partial Derivatives9m

Concluding Remarks2m

Do You Know Matrix Algebra?

Practice: Eigenvalues of a Symmetric Matrix10m

Practice: Distinct Real Eigenvalues10m

Practice: Complex-Conjugate Eigenvalues10m

Practice: Coupled Oscillators10m

Practice: Normal Modes of Coupled Oscillators10m

Practice: Fourier Series10m

Practice: Fourier series at x=010m

Practice: Fourier Series of a Square Wave10m

Do You Know Partial Derivatives?10m

Practice: Nondimensionalization of the Diffusion Equation10m

Practice: Boundary Conditions with Closed Pipe Ends10m

Practice: ODE Eigenvalue Problems10m

Practice: Solution of the Diffusion Equation with Closed Pipe Ends10m

Practice: Concentration of a Dye in a Pipe with Closed Ends10m

Please Rate this Course5m

Acknowledgements

Systems of Differential Equations20m

Normal Modes30m

Fourier Series30m

Separable Partial Differential Equations20m

The Diffusion Equation20m

Week Fours

HKUST - A dynamic, international research university, in relentless pursuit of excellence, leading the advance of science and technology, and educating the new generation of front-runners for Asia and the world....

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