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Beginner Level

Knowledge of single variable calculus.

Approx. 10 hours to complete

Suggested: 5 hours per week...

English

Subtitles: English

100% online

Start instantly and learn at your own schedule.

Beginner Level

Knowledge of single variable calculus.

Approx. 10 hours to complete

Suggested: 5 hours per week...

English

Subtitles: English

Syllabus - What you will learn from this course

Week
1
6 hours to complete

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 videos (Total 97 min), 11 readings, 6 quizzes
12 videos
Course Overview2m
Introduction to Differential Equations9m
Week 1 Introduction47s
Euler Method9m
Separable First-order Equations8m
Separable First-order Equation: Example6m
Linear First-order Equations13m
Linear First-order Equation: Example5m
Application: Compound Interest13m
Application: Terminal Velocity11m
Application: RC Circuit11m
Welcome and Course Information2m
Practice: Runge-Kutta Methods10m
Practice: Separable First-order Equations10m
Practice: Separable First-order Equation Examples10m
Practice: Linear First-order Equations5m
A Change of Variables Can Convert a Nonlinear Equation to a Linear equation10m
Practice: Linear First-order Equation: Examples10m
Practice: Compound Interest10m
Practice: Terminal Velocity10m
Practice: RC Circuit10m
6 practice exercises
Diagnostic Quiz15m
Classify Differential Equations10m
Separable First-order ODEs15m
Linear First-order ODEs15m
Applications20m
Week One1h
Week
2
8 hours to complete

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 videos (Total 218 min), 20 readings, 3 quizzes
22 videos
Euler Method for Higher-order ODEs9m
The Principle of Superposition6m
The Wronskian8m
Homogeneous Second-order ODE with Constant Coefficients9m
Case 1: Distinct Real Roots7m
Case 2: Complex-Conjugate Roots (Part A)7m
Case 2: Complex-Conjugate Roots (Part B)8m
Case 3: Repeated Roots (Part A)12m
Case 3: Repeated Roots (Part B)4m
Inhomogeneous Second-order ODE9m
Inhomogeneous Term: Exponential Function11m
Inhomogeneous Term: Sine or Cosine (Part A)9m
Inhomogeneous Term: Sine or Cosine (Part B)8m
Inhomogeneous Term: Polynomials7m
Resonance13m
RLC Circuit11m
Mass on a Spring9m
Pendulum12m
Damped Resonance14m
Complex Numbers17m
Nondimensionalization17m
Practice: Second-order Equation as System of First-order Equations10m
Practice: Second-order Runge-Kutta Method10m
Practice: Linear Superposition for Inhomogeneous ODEs10m
Practice: Wronskian of Exponential Function10m
Do You Know Complex Numbers?
Practice: Roots of the Characteristic Equation10m
Practice: Distinct Real Roots10m
Practice: Hyperbolic Sine and Cosine Functions10m
Practice: Complex-Conjugate Roots10m
Practice: Sine and Cosine Functions10m
Practice: Repeated Roots10m
Practice: Multiple Inhomogeneous Terms10m
Practice: Exponential Inhomogeneous Term10m
Practice: Sine or Cosine Inhomogeneous Term10m
Practice: Polynomial Inhomogeneous Term10m
When the Inhomogeneous Term is a Solution of the Homogeneous Equation10m
Do You Know Dimensional Analysis?
Another Nondimensionalization of the RLC Circuit Equation10m
Another Nondimensionalization of the Mass on a Spring Equation10m
Find the Amplitude of Oscillation10m
3 practice exercises
Homogeneous Equations20m
Inhomogeneous Equations20m
Week Two1h
Week
3
6 hours to complete

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 videos (Total 123 min), 10 readings, 4 quizzes
11 videos
Definition of the Laplace Transform13m
Laplace Transform of a Constant Coefficient ODE11m
Solution of an Initial Value Problem13m
The Heaviside Step Function10m
The Dirac Delta Function12m
Solution of a Discontinuous Inhomogeneous Term13m
Solution of an Impulsive Inhomogeneous Term7m
The Series Solution Method17m
Series Solution of the Airy's Equation (Part A)14m
Series Solution of the Airy's Equation (Part B)7m
Practice: The Laplace Transform of Sine10m
Practice: Laplace Transform of an ODE10m
Practice: Solution of an Initial Value Problem10m
Practice: Heaviside Step Function10m
Practice: The Dirac Delta Function15m
Practice: Discontinuous Inhomogeneous Term20m
Practice: Impulsive Inhomogeneous Term10m
Practice: Series Solution Method10m
Practice: Series Solution of a Nonconstant Coefficient ODE1m
Practice: Solution of the Airy's Equation10m
4 practice exercises
The Laplace Transform Method30m
Discontinuous and Impulsive Inhomogeneous Terms20m
Series Solutions20m
Week Three1h
Week
4
8 hours to complete

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 videos (Total 177 min), 17 readings, 6 quizzes
19 videos
Systems of Homogeneous Linear First-order ODEs8m
Distinct Real Eigenvalues9m
Complex-Conjugate Eigenvalues12m
Coupled Oscillators9m
Normal Modes (Eigenvalues)10m
Normal Modes (Eigenvectors)9m
Fourier Series12m
Fourier Sine and Cosine Series5m
Fourier Series: Example11m
The Diffusion Equation9m
Solution of the Diffusion Equation: Separation of Variables11m
Solution of the Diffusion Equation: Eigenvalues10m
Solution of the Diffusion Equation: Fourier Series9m
Diffusion Equation: Example10m
Matrices and Determinants13m
Eigenvalues and Eigenvectors10m
Partial Derivatives9m
Concluding Remarks2m
Do You Know Matrix Algebra?
Practice: Eigenvalues of a Symmetric Matrix10m
Practice: Distinct Real Eigenvalues10m
Practice: Complex-Conjugate Eigenvalues10m
Practice: Coupled Oscillators10m
Practice: Normal Modes of Coupled Oscillators10m
Practice: Fourier Series10m
Practice: Fourier series at x=010m
Practice: Fourier Series of a Square Wave10m
Do You Know Partial Derivatives?10m
Practice: Nondimensionalization of the Diffusion Equation10m
Practice: Boundary Conditions with Closed Pipe Ends10m
Practice: ODE Eigenvalue Problems10m
Practice: Solution of the Diffusion Equation with Closed Pipe Ends10m
Practice: Concentration of a Dye in a Pipe with Closed Ends10m
Acknowledgements
6 practice exercises
Systems of Differential Equations20m
Normal Modes30m
Fourier Series30m
Separable Partial Differential Equations20m
The Diffusion Equation20m
Week Four1h
4.8
16 Reviews

Top Reviews

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!

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