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
Differential Equations for Engineers
proposé par
Université des sciences et technologies de Hong Kong
Programme du cours : ce que vous apprendrez dans ce cours
Semaine
16 heures pour terminer
First-Order Differential Equations
Welcome 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 vidéos (Total 97 min), 11 lectures, 6 quiz
12 vidéos
Course Trailer4 min
Course Overview2 min
Introduction to Differential Equations9 min
Week 1 Introduction47s
Euler Method9 min
Separable First-order Equations8 min
Separable First-order Equation: Example6 min
Linear First-order Equations13 min
Linear First-order Equation: Example5 min
Application: Compound Interest13 min
Application: Terminal Velocity11 min
Application: RC Circuit11 min
11 lectures
Welcome and Course Information2 min
Get to Know Your Classmates10 min
Practice: Runge-Kutta Methods10 min
Practice: Separable First-order Equations10 min
Practice: Separable First-order Equation Examples10 min
Practice: Linear First-order Equations5 min
A Change of Variables Can Convert a Nonlinear Equation to a Linear equation10 min
Practice: Linear First-order Equation: Examples10 min
Practice: Compound Interest10 min
Practice: Terminal Velocity10 min
Practice: RC Circuit10 min
6 exercices pour s'entraîner
Diagnostic Quiz15 min
Classify Differential Equations10 min
Separable First-order ODEs15 min
Linear First-order ODEs15 min
Applications20 min
Week Ones
Semaine
28 heures pour terminer
Second-Order Differential Equations
We 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 vidéos (Total 218 min), 20 lectures, 3 quiz
22 vidéos
Week 2 Introduction1 min
Euler Method for Higher-order ODEs9 min
The Principle of Superposition6 min
The Wronskian8 min
Homogeneous Second-order ODE with Constant Coefficients9 min
Case 1: Distinct Real Roots7 min
Case 2: Complex-Conjugate Roots (Part A)7 min
Case 2: Complex-Conjugate Roots (Part B)8 min
Case 3: Repeated Roots (Part A)12 min
Case 3: Repeated Roots (Part B)4 min
Inhomogeneous Second-order ODE9 min
Inhomogeneous Term: Exponential Function11 min
Inhomogeneous Term: Sine or Cosine (Part A)9 min
Inhomogeneous Term: Sine or Cosine (Part B)8 min
Inhomogeneous Term: Polynomials7 min
Resonance13 min
RLC Circuit11 min
Mass on a Spring9 min
Pendulum12 min
Damped Resonance14 min
Complex Numbers17 min
Nondimensionalization17 min
20 lectures
Practice: Second-order Equation as System of First-order Equations10 min
Practice: Second-order Runge-Kutta Method10 min
Practice: Linear Superposition for Inhomogeneous ODEs10 min
Practice: Wronskian of Exponential Function10 min
Do You Know Complex Numbers?
Practice: Roots of the Characteristic Equation10 min
Practice: Distinct Real Roots10 min
Practice: Hyperbolic Sine and Cosine Functions10 min
Practice: Complex-Conjugate Roots10 min
Practice: Sine and Cosine Functions10 min
Practice: Repeated Roots10 min
Practice: Multiple Inhomogeneous Terms10 min
Practice: Exponential Inhomogeneous Term10 min
Practice: Sine or Cosine Inhomogeneous Term10 min
Practice: Polynomial Inhomogeneous Term10 min
When the Inhomogeneous Term is a Solution of the Homogeneous Equation10 min
Do You Know Dimensional Analysis?
Another Nondimensionalization of the RLC Circuit Equation10 min
Another Nondimensionalization of the Mass on a Spring Equation10 min
Find the Amplitude of Oscillation10 min
3 exercices pour s'entraîner
Homogeneous Equations20 min
Inhomogeneous Equations20 min
Week Twos
Semaine
36 heures pour terminer
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 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 vidéos (Total 123 min), 10 lectures, 4 quiz
11 vidéos
Definition of the Laplace Transform13 min
Laplace Transform of a Constant Coefficient ODE11 min
Solution of an Initial Value Problem13 min
The Heaviside Step Function10 min
The Dirac Delta Function12 min
Solution of a Discontinuous Inhomogeneous Term13 min
Solution of an Impulsive Inhomogeneous Term7 min
The Series Solution Method17 min
Series Solution of the Airy's Equation (Part A)14 min
Series Solution of the Airy's Equation (Part B)7 min
10 lectures
Practice: The Laplace Transform of Sine10 min
Practice: Laplace Transform of an ODE10 min
Practice: Solution of an Initial Value Problem10 min
Practice: Heaviside Step Function10 min
Practice: The Dirac Delta Function15 min
Practice: Discontinuous Inhomogeneous Term20 min
Practice: Impulsive Inhomogeneous Term10 min
Practice: Series Solution Method10 min
Practice: Series Solution of a Nonconstant Coefficient ODE1 min
Practice: Solution of the Airy's Equation10 min
4 exercices pour s'entraîner
The Laplace Transform Method30 min
Discontinuous and Impulsive Inhomogeneous Terms20 min
Series Solutions20 min
Week Threes
Semaine
48 heures pour terminer
Systems of Differential Equations and Partial Differential Equations
We 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 vidéos (Total 177 min), 17 lectures, 6 quiz
19 vidéos
Systems of Homogeneous Linear First-order ODEs8 min
Distinct Real Eigenvalues9 min
Complex-Conjugate Eigenvalues12 min
Coupled Oscillators9 min
Normal Modes (Eigenvalues)10 min
Normal Modes (Eigenvectors)9 min
Fourier Series12 min
Fourier Sine and Cosine Series5 min
Fourier Series: Example11 min
The Diffusion Equation9 min
Solution of the Diffusion Equation: Separation of Variables11 min
Solution of the Diffusion Equation: Eigenvalues10 min
Solution of the Diffusion Equation: Fourier Series9 min
Diffusion Equation: Example10 min
Matrices and Determinants13 min
Eigenvalues and Eigenvectors10 min
Partial Derivatives9 min
Concluding Remarks2 min
17 lectures
Do You Know Matrix Algebra?
Practice: Eigenvalues of a Symmetric Matrix10 min
Practice: Distinct Real Eigenvalues10 min
Practice: Complex-Conjugate Eigenvalues10 min
Practice: Coupled Oscillators10 min
Practice: Normal Modes of Coupled Oscillators10 min
Practice: Fourier Series10 min
Practice: Fourier series at x=010 min
Practice: Fourier Series of a Square Wave10 min
Do You Know Partial Derivatives?10 min
Practice: Nondimensionalization of the Diffusion Equation10 min
Practice: Boundary Conditions with Closed Pipe Ends10 min
Practice: ODE Eigenvalue Problems10 min
Practice: Solution of the Diffusion Equation with Closed Pipe Ends10 min
Practice: Concentration of a Dye in a Pipe with Closed Ends10 min
Please Rate this Course5 min
Acknowledgements
6 exercices pour s'entraîner
Systems of Differential Equations20 min
Normal Modes30 min
Fourier Series30 min
Separable Partial Differential Equations20 min
The Diffusion Equation20 min
Week Fours
Enseignants
Jeffrey R. Chasnov
ProfessorDepartment of Mathematics
À propos de Université des sciences et technologies de Hong Kong
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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