4.8
8 个评分
1 个审阅

#### 初级

Knowledge of single variable calculus.

#### 初级

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 个视频 （总计 97 分钟）, 11 个阅读材料, 6 个测验
12 个视频
Course Overview2分钟
Introduction to Differential Equations9分钟
Week 1 Introduction47
Euler Method9分钟
Separable First-order Equations8分钟
Separable First-order Equation: Example6分钟
Linear First-order Equations13分钟
Linear First-order Equation: Example5分钟
Application: Compound Interest13分钟
Application: Terminal Velocity11分钟
Application: RC Circuit11分钟
11 个阅读材料
Welcome and Course Information2分钟
Get to Know Your Classmates10分钟
Practice: Runge-Kutta Methods10分钟
Practice: Separable First-order Equations10分钟
Practice: Separable First-order Equation Examples10分钟
Practice: Linear First-order Equations5分钟
A Change of Variables Can Convert a Nonlinear Equation to a Linear equation10分钟
Practice: Linear First-order Equation: Examples10分钟
Practice: Compound Interest10分钟
Practice: Terminal Velocity10分钟
Practice: RC Circuit10分钟
6 个练习
Diagnostic Quiz15分钟
Classify Differential Equations10分钟
Separable First-order ODEs15分钟
Linear First-order ODEs15分钟
Applications20分钟
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 个视频 （总计 218 分钟）, 20 个阅读材料, 3 个测验
22 个视频
Euler Method for Higher-order ODEs9分钟
The Principle of Superposition6分钟
The Wronskian8分钟
Homogeneous Second-order ODE with Constant Coefficients9分钟
Case 1: Distinct Real Roots7分钟
Case 2: Complex-Conjugate Roots (Part A)7分钟
Case 2: Complex-Conjugate Roots (Part B)8分钟
Case 3: Repeated Roots (Part A)12分钟
Case 3: Repeated Roots (Part B)4分钟
Inhomogeneous Second-order ODE9分钟
Inhomogeneous Term: Exponential Function11分钟
Inhomogeneous Term: Sine or Cosine (Part A)9分钟
Inhomogeneous Term: Sine or Cosine (Part B)8分钟
Inhomogeneous Term: Polynomials7分钟
Resonance13分钟
RLC Circuit11分钟
Mass on a Spring9分钟
Pendulum12分钟
Damped Resonance14分钟
Complex Numbers17分钟
Nondimensionalization17分钟
20 个阅读材料
Practice: Second-order Equation as System of First-order Equations10分钟
Practice: Second-order Runge-Kutta Method10分钟
Practice: Linear Superposition for Inhomogeneous ODEs10分钟
Practice: Wronskian of Exponential Function10分钟
Do You Know Complex Numbers?
Practice: Roots of the Characteristic Equation10分钟
Practice: Distinct Real Roots10分钟
Practice: Hyperbolic Sine and Cosine Functions10分钟
Practice: Complex-Conjugate Roots10分钟
Practice: Sine and Cosine Functions10分钟
Practice: Repeated Roots10分钟
Practice: Multiple Inhomogeneous Terms10分钟
Practice: Exponential Inhomogeneous Term10分钟
Practice: Sine or Cosine Inhomogeneous Term10分钟
Practice: Polynomial Inhomogeneous Term10分钟
When the Inhomogeneous Term is a Solution of the Homogeneous Equation10分钟
Do You Know Dimensional Analysis?
Another Nondimensionalization of the RLC Circuit Equation10分钟
Another Nondimensionalization of the Mass on a Spring Equation10分钟
Find the Amplitude of Oscillation10分钟
3 个练习
Homogeneous Equations20分钟
Inhomogeneous Equations20分钟
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 个视频 （总计 123 分钟）, 10 个阅读材料, 4 个测验
11 个视频
Definition of the Laplace Transform13分钟
Laplace Transform of a Constant Coefficient ODE11分钟
Solution of an Initial Value Problem13分钟
The Heaviside Step Function10分钟
The Dirac Delta Function12分钟
Solution of a Discontinuous Inhomogeneous Term13分钟
Solution of an Impulsive Inhomogeneous Term7分钟
The Series Solution Method17分钟
Series Solution of the Airy's Equation (Part A)14分钟
Series Solution of the Airy's Equation (Part B)7分钟
10 个阅读材料
Practice: The Laplace Transform of Sine10分钟
Practice: Laplace Transform of an ODE10分钟
Practice: Solution of an Initial Value Problem10分钟
Practice: Heaviside Step Function10分钟
Practice: The Dirac Delta Function15分钟
Practice: Discontinuous Inhomogeneous Term20分钟
Practice: Impulsive Inhomogeneous Term10分钟
Practice: Series Solution Method10分钟
Practice: Series Solution of a Nonconstant Coefficient ODE1分钟
Practice: Solution of the Airy's Equation10分钟
4 个练习
The Laplace Transform Method30分钟
Discontinuous and Impulsive Inhomogeneous Terms20分钟
Series Solutions20分钟
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 个视频 （总计 177 分钟）, 17 个阅读材料, 6 个测验
19 个视频
Systems of Homogeneous Linear First-order ODEs8分钟
Distinct Real Eigenvalues9分钟
Complex-Conjugate Eigenvalues12分钟
Coupled Oscillators9分钟
Normal Modes (Eigenvalues)10分钟
Normal Modes (Eigenvectors)9分钟
Fourier Series12分钟
Fourier Sine and Cosine Series5分钟
Fourier Series: Example11分钟
The Diffusion Equation9分钟
Solution of the Diffusion Equation: Separation of Variables11分钟
Solution of the Diffusion Equation: Eigenvalues10分钟
Solution of the Diffusion Equation: Fourier Series9分钟
Diffusion Equation: Example10分钟
Matrices and Determinants13分钟
Eigenvalues and Eigenvectors10分钟
Partial Derivatives9分钟
Concluding Remarks2分钟
17 个阅读材料
Do You Know Matrix Algebra?
Practice: Eigenvalues of a Symmetric Matrix10分钟
Practice: Distinct Real Eigenvalues10分钟
Practice: Complex-Conjugate Eigenvalues10分钟
Practice: Coupled Oscillators10分钟
Practice: Normal Modes of Coupled Oscillators10分钟
Practice: Fourier Series10分钟
Practice: Fourier series at x=010分钟
Practice: Fourier Series of a Square Wave10分钟
Do You Know Partial Derivatives?10分钟
Practice: Nondimensionalization of the Diffusion Equation10分钟
Practice: Boundary Conditions with Closed Pipe Ends10分钟
Practice: ODE Eigenvalue Problems10分钟
Practice: Solution of the Diffusion Equation with Closed Pipe Ends10分钟
Practice: Concentration of a Dye in a Pipe with Closed Ends10分钟
Please Rate this Course5分钟
Acknowledgements
6 个练习
Systems of Differential Equations20分钟
Normal Modes30分钟
Fourier Series30分钟
Separable Partial Differential Equations20分钟
The Diffusion Equation20分钟
Week Four

## 关于 香港科技大学

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