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#### 初級

Knowledge of single variable calculus.

#### 初級

Knowledge of single variable calculus.

1

## 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 個視頻 （總計 97 分鐘）, 11 個閱讀材料, 6 個測驗
12 個視頻
Course Overview2分鐘
Introduction to Differential Equations9分鐘
Week 1 Introduction47
Euler Method9分鐘
Separable First-order Equations8分鐘
Separable First-order Equation: Example6分鐘
Linear First-order Equations13分鐘
Linear First-order Equation: Example5分鐘
Application: Compound Interest13分鐘
Application: Terminal Velocity11分鐘
Application: RC Circuit11分鐘
11 個閱讀材料
Welcome and Course Information2分鐘
Practice: Runge-Kutta Methods10分鐘
Practice: Separable First-order Equations10分鐘
Practice: Separable First-order Equation Examples10分鐘
Practice: Linear First-order Equations5分鐘
A Change of Variables Can Convert a Nonlinear Equation to a Linear equation10分鐘
Practice: Linear First-order Equation: Examples10分鐘
Practice: Compound Interest10分鐘
Practice: Terminal Velocity10分鐘
Practice: RC Circuit10分鐘
6 個練習
Diagnostic Quiz15分鐘
Classify Differential Equations10分鐘
Separable First-order ODEs15分鐘
Linear First-order ODEs15分鐘
Applications20分鐘
Week One
2

## 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 個視頻 （總計 218 分鐘）, 20 個閱讀材料, 3 個測驗
22 個視頻
Euler Method for Higher-order ODEs9分鐘
The Principle of Superposition6分鐘
The Wronskian8分鐘
Homogeneous Second-order ODE with Constant Coefficients9分鐘
Case 1: Distinct Real Roots7分鐘
Case 2: Complex-Conjugate Roots (Part A)7分鐘
Case 2: Complex-Conjugate Roots (Part B)8分鐘
Case 3: Repeated Roots (Part A)12分鐘
Case 3: Repeated Roots (Part B)4分鐘
Inhomogeneous Second-order ODE9分鐘
Inhomogeneous Term: Exponential Function11分鐘
Inhomogeneous Term: Sine or Cosine (Part A)9分鐘
Inhomogeneous Term: Sine or Cosine (Part B)8分鐘
Inhomogeneous Term: Polynomials7分鐘
Resonance13分鐘
RLC Circuit11分鐘
Mass on a Spring9分鐘
Pendulum12分鐘
Damped Resonance14分鐘
Complex Numbers17分鐘
Nondimensionalization17分鐘
20 個閱讀材料
Practice: Second-order Equation as System of First-order Equations10分鐘
Practice: Second-order Runge-Kutta Method10分鐘
Practice: Linear Superposition for Inhomogeneous ODEs10分鐘
Practice: Wronskian of Exponential Function10分鐘
Do You Know Complex Numbers?
Practice: Roots of the Characteristic Equation10分鐘
Practice: Distinct Real Roots10分鐘
Practice: Hyperbolic Sine and Cosine Functions10分鐘
Practice: Complex-Conjugate Roots10分鐘
Practice: Sine and Cosine Functions10分鐘
Practice: Repeated Roots10分鐘
Practice: Multiple Inhomogeneous Terms10分鐘
Practice: Exponential Inhomogeneous Term10分鐘
Practice: Sine or Cosine Inhomogeneous Term10分鐘
Practice: Polynomial Inhomogeneous Term10分鐘
When the Inhomogeneous Term is a Solution of the Homogeneous Equation10分鐘
Do You Know Dimensional Analysis?
Another Nondimensionalization of the RLC Circuit Equation10分鐘
Another Nondimensionalization of the Mass on a Spring Equation10分鐘
Find the Amplitude of Oscillation10分鐘
3 個練習
Homogeneous Equations20分鐘
Inhomogeneous Equations20分鐘
Week Two
3

## 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 個視頻 （總計 123 分鐘）, 10 個閱讀材料, 4 個測驗
11 個視頻
Definition of the Laplace Transform13分鐘
Laplace Transform of a Constant Coefficient ODE11分鐘
Solution of an Initial Value Problem13分鐘
The Heaviside Step Function10分鐘
The Dirac Delta Function12分鐘
Solution of a Discontinuous Inhomogeneous Term13分鐘
Solution of an Impulsive Inhomogeneous Term7分鐘
The Series Solution Method17分鐘
Series Solution of the Airy's Equation (Part A)14分鐘
Series Solution of the Airy's Equation (Part B)7分鐘
10 個閱讀材料
Practice: The Laplace Transform of Sine10分鐘
Practice: Laplace Transform of an ODE10分鐘
Practice: Solution of an Initial Value Problem10分鐘
Practice: Heaviside Step Function10分鐘
Practice: The Dirac Delta Function15分鐘
Practice: Discontinuous Inhomogeneous Term20分鐘
Practice: Impulsive Inhomogeneous Term10分鐘
Practice: Series Solution Method10分鐘
Practice: Series Solution of a Nonconstant Coefficient ODE1分鐘
Practice: Solution of the Airy's Equation10分鐘
4 個練習
The Laplace Transform Method30分鐘
Discontinuous and Impulsive Inhomogeneous Terms20分鐘
Series Solutions20分鐘
Week Three
4

## 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 個視頻 （總計 177 分鐘）, 17 個閱讀材料, 6 個測驗
19 個視頻
Systems of Homogeneous Linear First-order ODEs8分鐘
Distinct Real Eigenvalues9分鐘
Complex-Conjugate Eigenvalues12分鐘
Coupled Oscillators9分鐘
Normal Modes (Eigenvalues)10分鐘
Normal Modes (Eigenvectors)9分鐘
Fourier Series12分鐘
Fourier Sine and Cosine Series5分鐘
Fourier Series: Example11分鐘
The Diffusion Equation9分鐘
Solution of the Diffusion Equation: Separation of Variables11分鐘
Solution of the Diffusion Equation: Eigenvalues10分鐘
Solution of the Diffusion Equation: Fourier Series9分鐘
Diffusion Equation: Example10分鐘
Matrices and Determinants13分鐘
Eigenvalues and Eigenvectors10分鐘
Partial Derivatives9分鐘
Concluding Remarks2分鐘
17 個閱讀材料
Do You Know Matrix Algebra?
Practice: Eigenvalues of a Symmetric Matrix10分鐘
Practice: Distinct Real Eigenvalues10分鐘
Practice: Complex-Conjugate Eigenvalues10分鐘
Practice: Coupled Oscillators10分鐘
Practice: Normal Modes of Coupled Oscillators10分鐘
Practice: Fourier Series10分鐘
Practice: Fourier series at x=010分鐘
Practice: Fourier Series of a Square Wave10分鐘
Do You Know Partial Derivatives?10分鐘
Practice: Nondimensionalization of the Diffusion Equation10分鐘
Practice: Boundary Conditions with Closed Pipe Ends10分鐘
Practice: ODE Eigenvalue Problems10分鐘
Practice: Solution of the Diffusion Equation with Closed Pipe Ends10分鐘
Practice: Concentration of a Dye in a Pipe with Closed Ends10分鐘
Acknowledgements
6 個練習
Systems of Differential Equations20分鐘
Normal Modes30分鐘
Fourier Series30分鐘
Separable Partial Differential Equations20分鐘
The Diffusion Equation20分鐘
Week Four
4.8
14 個審閱

### 熱門審閱

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

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

## 講師

### Jeffrey R. Chasnov

Professor
Department of Mathematics

## 關於 香港科技大学

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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