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Nível iniciante

Knowledge of single variable calculus.

Aprox. 11 horas para completar

Sugerido: 5 hours per week...

Inglês

Legendas: Inglês

100% online

Comece imediatamente e aprenda em seu próprio cronograma.

Prazos flexíveis

Redefinir os prazos de acordo com sua programação.

Nível iniciante

Knowledge of single variable calculus.

Aprox. 11 horas para completar

Sugerido: 5 hours per week...

Inglês

Legendas: Inglês

Programa - O que você aprenderá com este curso

Semana
1
6 horas para concluir

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 vídeos (total de (Total 97 mín.) min), 11 leituras, 6 testes
12 videos
Course Overview2min
Introduction to Differential Equations9min
Week 1 Introduction47s
Euler Method9min
Separable First-order Equations8min
Separable First-order Equation: Example6min
Linear First-order Equations13min
Linear First-order Equation: Example5min
Application: Compound Interest13min
Application: Terminal Velocity11min
Application: RC Circuit11min
11 leituras
Welcome and Course Information2min
Get to Know Your Classmates10min
Practice: Runge-Kutta Methods10min
Practice: Separable First-order Equations10min
Practice: Separable First-order Equation Examples10min
Practice: Linear First-order Equations5min
A Change of Variables Can Convert a Nonlinear Equation to a Linear equation10min
Practice: Linear First-order Equation: Examples10min
Practice: Compound Interest10min
Practice: Terminal Velocity10min
Practice: RC Circuit10min
6 exercícios práticos
Diagnostic Quiz15min
Classify Differential Equations10min
Separable First-order ODEs15min
Linear First-order ODEs15min
Applications20min
Week Ones
Semana
2
8 horas para concluir

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 vídeos (total de (Total 218 mín.) min), 20 leituras, 3 testes
22 videos
Euler Method for Higher-order ODEs9min
The Principle of Superposition6min
The Wronskian8min
Homogeneous Second-order ODE with Constant Coefficients9min
Case 1: Distinct Real Roots7min
Case 2: Complex-Conjugate Roots (Part A)7min
Case 2: Complex-Conjugate Roots (Part B)8min
Case 3: Repeated Roots (Part A)12min
Case 3: Repeated Roots (Part B)4min
Inhomogeneous Second-order ODE9min
Inhomogeneous Term: Exponential Function11min
Inhomogeneous Term: Sine or Cosine (Part A)9min
Inhomogeneous Term: Sine or Cosine (Part B)8min
Inhomogeneous Term: Polynomials7min
Resonance13min
RLC Circuit11min
Mass on a Spring9min
Pendulum12min
Damped Resonance14min
Complex Numbers17min
Nondimensionalization17min
20 leituras
Practice: Second-order Equation as System of First-order Equations10min
Practice: Second-order Runge-Kutta Method10min
Practice: Linear Superposition for Inhomogeneous ODEs10min
Practice: Wronskian of Exponential Function10min
Do You Know Complex Numbers?
Practice: Roots of the Characteristic Equation10min
Practice: Distinct Real Roots10min
Practice: Hyperbolic Sine and Cosine Functions10min
Practice: Complex-Conjugate Roots10min
Practice: Sine and Cosine Functions10min
Practice: Repeated Roots10min
Practice: Multiple Inhomogeneous Terms10min
Practice: Exponential Inhomogeneous Term10min
Practice: Sine or Cosine Inhomogeneous Term10min
Practice: Polynomial Inhomogeneous Term10min
When the Inhomogeneous Term is a Solution of the Homogeneous Equation10min
Do You Know Dimensional Analysis?
Another Nondimensionalization of the RLC Circuit Equation10min
Another Nondimensionalization of the Mass on a Spring Equation10min
Find the Amplitude of Oscillation10min
3 exercícios práticos
Homogeneous Equations20min
Inhomogeneous Equations20min
Week Twos
Semana
3
6 horas para concluir

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 vídeos (total de (Total 123 mín.) min), 10 leituras, 4 testes
11 videos
Definition of the Laplace Transform13min
Laplace Transform of a Constant Coefficient ODE11min
Solution of an Initial Value Problem13min
The Heaviside Step Function10min
The Dirac Delta Function12min
Solution of a Discontinuous Inhomogeneous Term13min
Solution of an Impulsive Inhomogeneous Term7min
The Series Solution Method17min
Series Solution of the Airy's Equation (Part A)14min
Series Solution of the Airy's Equation (Part B)7min
10 leituras
Practice: The Laplace Transform of Sine10min
Practice: Laplace Transform of an ODE10min
Practice: Solution of an Initial Value Problem10min
Practice: Heaviside Step Function10min
Practice: The Dirac Delta Function15min
Practice: Discontinuous Inhomogeneous Term20min
Practice: Impulsive Inhomogeneous Term10min
Practice: Series Solution Method10min
Practice: Series Solution of a Nonconstant Coefficient ODE1min
Practice: Solution of the Airy's Equation10min
4 exercícios práticos
The Laplace Transform Method30min
Discontinuous and Impulsive Inhomogeneous Terms20min
Series Solutions20min
Week Threes
Semana
4
8 horas para concluir

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 vídeos (total de (Total 177 mín.) min), 17 leituras, 6 testes
19 videos
Systems of Homogeneous Linear First-order ODEs8min
Distinct Real Eigenvalues9min
Complex-Conjugate Eigenvalues12min
Coupled Oscillators9min
Normal Modes (Eigenvalues)10min
Normal Modes (Eigenvectors)9min
Fourier Series12min
Fourier Sine and Cosine Series5min
Fourier Series: Example11min
The Diffusion Equation9min
Solution of the Diffusion Equation: Separation of Variables11min
Solution of the Diffusion Equation: Eigenvalues10min
Solution of the Diffusion Equation: Fourier Series9min
Diffusion Equation: Example10min
Matrices and Determinants13min
Eigenvalues and Eigenvectors10min
Partial Derivatives9min
Concluding Remarks2min
17 leituras
Do You Know Matrix Algebra?
Practice: Eigenvalues of a Symmetric Matrix10min
Practice: Distinct Real Eigenvalues10min
Practice: Complex-Conjugate Eigenvalues10min
Practice: Coupled Oscillators10min
Practice: Normal Modes of Coupled Oscillators10min
Practice: Fourier Series10min
Practice: Fourier series at x=010min
Practice: Fourier Series of a Square Wave10min
Do You Know Partial Derivatives?10min
Practice: Nondimensionalization of the Diffusion Equation10min
Practice: Boundary Conditions with Closed Pipe Ends10min
Practice: ODE Eigenvalue Problems10min
Practice: Solution of the Diffusion Equation with Closed Pipe Ends10min
Practice: Concentration of a Dye in a Pipe with Closed Ends10min
Please Rate this Course5min
Acknowledgements
6 exercícios práticos
Systems of Differential Equations20min
Normal Modes30min
Fourier Series30min
Separable Partial Differential Equations20min
The Diffusion Equation20min
Week Fours

Instrutores

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Jeffrey R. Chasnov

Professor
Department of Mathematics

Sobre Universidade de Ciência e Tecnologia 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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