このコースについて
25,100 最近の表示

100%オンライン

自分のスケジュールですぐに学習を始めてください。

柔軟性のある期限

スケジュールに従って期限をリセットします。

初級レベル

Knowledge of single variable calculus.

約10時間で修了

推奨:5 hours per week...

英語

字幕:英語

100%オンライン

自分のスケジュールですぐに学習を始めてください。

柔軟性のある期限

スケジュールに従って期限をリセットします。

初級レベル

Knowledge of single variable calculus.

約10時間で修了

推奨:5 hours per week...

英語

字幕:英語

シラバス - 本コースの学習内容

1
6時間で修了

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件のビデオ (合計97分), 11 readings, 6 quizzes
12件のビデオ
Week 1 Introduction47
Euler Method9 分
Separable First-order Equations8 分
Separable First-order Equation: Example6 分
Linear First-order Equations13 分
Linear First-order Equation: Example5 分
Application: Compound Interest13 分
Application: Terminal Velocity11 分
Application: RC Circuit11 分
11件の学習用教材
Welcome and Course Information2 分
Get to Know Your Classmates10 分
Practice: Runge-Kutta Methods10 分
Practice: Separable First-order Equations10 分
Practice: Separable First-order Equation Examples10 分
Practice: Linear First-order Equations5 分
A Change of Variables Can Convert a Nonlinear Equation to a Linear equation10 分
Practice: Linear First-order Equation: Examples10 分
Practice: Compound Interest10 分
Practice: Terminal Velocity10 分
Practice: RC Circuit10 分
6の練習問題
Diagnostic Quiz15 分
Classify Differential Equations10 分
Separable First-order ODEs15 分
Linear First-order ODEs15 分
Applications20 分
Week One1 時間
2
8時間で修了

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件のビデオ (合計218分), 20 readings, 3 quizzes
22件のビデオ
The Wronskian8 分
Homogeneous Second-order ODE with Constant Coefficients9 分
Case 1: Distinct Real Roots7 分
Case 2: Complex-Conjugate Roots (Part A)7 分
Case 2: Complex-Conjugate Roots (Part B)8 分
Case 3: Repeated Roots (Part A)12 分
Case 3: Repeated Roots (Part B)4 分
Inhomogeneous Second-order ODE9 分
Inhomogeneous Term: Exponential Function11 分
Inhomogeneous Term: Sine or Cosine (Part A)9 分
Inhomogeneous Term: Sine or Cosine (Part B)8 分
Inhomogeneous Term: Polynomials7 分
Resonance13 分
RLC Circuit11 分
Mass on a Spring9 分
Pendulum12 分
Damped Resonance14 分
Complex Numbers17 分
Nondimensionalization17 分
20件の学習用教材
Practice: Second-order Equation as System of First-order Equations10 分
Practice: Second-order Runge-Kutta Method10 分
Practice: Linear Superposition for Inhomogeneous ODEs10 分
Practice: Wronskian of Exponential Function10 分
Do You Know Complex Numbers?
Practice: Roots of the Characteristic Equation10 分
Practice: Distinct Real Roots10 分
Practice: Hyperbolic Sine and Cosine Functions10 分
Practice: Complex-Conjugate Roots10 分
Practice: Sine and Cosine Functions10 分
Practice: Repeated Roots10 分
Practice: Multiple Inhomogeneous Terms10 分
Practice: Exponential Inhomogeneous Term10 分
Practice: Sine or Cosine Inhomogeneous Term10 分
Practice: Polynomial Inhomogeneous Term10 分
When the Inhomogeneous Term is a Solution of the Homogeneous Equation10 分
Do You Know Dimensional Analysis?
Another Nondimensionalization of the RLC Circuit Equation10 分
Another Nondimensionalization of the Mass on a Spring Equation10 分
Find the Amplitude of Oscillation10 分
3の練習問題
Homogeneous Equations20 分
Inhomogeneous Equations20 分
Week Two1 時間
3
6時間で修了

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件のビデオ (合計123分), 10 readings, 4 quizzes
11件のビデオ
Solution of an Initial Value Problem13 分
The Heaviside Step Function10 分
The Dirac Delta Function12 分
Solution of a Discontinuous Inhomogeneous Term13 分
Solution of an Impulsive Inhomogeneous Term7 分
The Series Solution Method17 分
Series Solution of the Airy's Equation (Part A)14 分
Series Solution of the Airy's Equation (Part B)7 分
10件の学習用教材
Practice: The Laplace Transform of Sine10 分
Practice: Laplace Transform of an ODE10 分
Practice: Solution of an Initial Value Problem10 分
Practice: Heaviside Step Function10 分
Practice: The Dirac Delta Function15 分
Practice: Discontinuous Inhomogeneous Term20 分
Practice: Impulsive Inhomogeneous Term10 分
Practice: Series Solution Method10 分
Practice: Series Solution of a Nonconstant Coefficient ODE1 分
Practice: Solution of the Airy's Equation10 分
4の練習問題
The Laplace Transform Method30 分
Discontinuous and Impulsive Inhomogeneous Terms20 分
Series Solutions20 分
Week Three1 時間
4
8時間で修了

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件のビデオ (合計177分), 17 readings, 6 quizzes
19件のビデオ
Complex-Conjugate Eigenvalues12 分
Coupled Oscillators9 分
Normal Modes (Eigenvalues)10 分
Normal Modes (Eigenvectors)9 分
Fourier Series12 分
Fourier Sine and Cosine Series5 分
Fourier Series: Example11 分
The Diffusion Equation9 分
Solution of the Diffusion Equation: Separation of Variables11 分
Solution of the Diffusion Equation: Eigenvalues10 分
Solution of the Diffusion Equation: Fourier Series9 分
Diffusion Equation: Example10 分
Matrices and Determinants13 分
Eigenvalues and Eigenvectors10 分
Partial Derivatives9 分
Concluding Remarks2 分
17件の学習用教材
Do You Know Matrix Algebra?
Practice: Eigenvalues of a Symmetric Matrix10 分
Practice: Distinct Real Eigenvalues10 分
Practice: Complex-Conjugate Eigenvalues10 分
Practice: Coupled Oscillators10 分
Practice: Normal Modes of Coupled Oscillators10 分
Practice: Fourier Series10 分
Practice: Fourier series at x=010 分
Practice: Fourier Series of a Square Wave10 分
Do You Know Partial Derivatives?10 分
Practice: Nondimensionalization of the Diffusion Equation10 分
Practice: Boundary Conditions with Closed Pipe Ends10 分
Practice: ODE Eigenvalue Problems10 分
Practice: Solution of the Diffusion Equation with Closed Pipe Ends10 分
Practice: Concentration of a Dye in a Pipe with Closed Ends10 分
Please Rate this Course5 分
Acknowledgements
6の練習問題
Systems of Differential Equations20 分
Normal Modes30 分
Fourier Series30 分
Separable Partial Differential Equations20 分
The Diffusion Equation20 分
Week Four1 時間
4.8
16件のレビューChevron Right

Differential Equations for Engineers からの人気レビュー

by YHApr 3rd 2019

Thank you Prof. Chasnov. The lectures are really impressive and explain derivations throughly. I cannot enjoy more on a math course than this one.

by SFMay 23rd 2019

I can't be thankful enough for this course. It was a life changing for me. Thank you VERY much!

講師

Avatar

Jeffrey R. Chasnov

Professor
Department of Mathematics

香港科技大学(The Hong Kong University of Science and Technology)について

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....

よくある質問

  • 修了証に登録すると、すべてのビデオ、テスト、およびプログラミング課題(該当する場合)にアクセスできます。ピアレビュー課題は、セッションが開始してからのみ、提出およびレビューできます。購入せずにコースを検討することを選択する場合、特定の課題にアクセスすることはできません。

  • 修了証を購入する際、コースのすべての教材(採点課題を含む)にアクセスできます。コースを完了すると、電子修了証が成果のページに追加されます。そこから修了証を印刷したり、LinkedInのプロフィールに追加したりできます。コースの内容の閲覧のみを希望する場合は、無料でコースを聴講できます。

さらに質問がある場合は、受講者向けヘルプセンターにアクセスしてください。