Differential Equations
This playlist contains 32.5 hours across 52 video lectures from my Differential Equations course (Math 2080) at Clemson University. After teaching this course 9 times during the academic year, I offered it online and made these videos. Every time I taught it, I would discover a better way to present the material until it finally "converged" to its present form, which I am very happy with. It's remarkable how much one can learn from teaching a course, even if one knows the material very well.
For many, if not most undergraduate STEM majors, Differential Equations is the most important math class they will take. Unfortunately, the way that it is usually taught lacks some of the key intution, and presents the material as a "cook book" of how to solve a variety of ODEs. As a result, students usually miss the "big picture", which is quite beautiful. In this course, I make a special effort to explain the simple intuition behind the scenes, because understanding that makes so much of the content much more straightforward and natural.
My presentation of the material does really not follow any textbook. Despite this, most textbooks (at least those "with boundary value problems") would serve as a decent reference, as they contain most of this content. That said, in this day and age, so does the internet.
I have a course webpage, where I make the content freely available. With each lecture, I post the slides and a worksheet containing what I consider to be the fundamental practice problems for that topic. It should be very conducive to self-study.
http://www.math.clemson.edu/~macaule/math2080-online.html
Can't get enough? Check out my course "Advanced Engineering Mathematics", which can be thought of as the natural follow-up to this course.
http://www.math.clemson.edu/~macaule/math4340-online.html
Curated by: Professor Macauley (52 videos)
Currently Playing: Differential Equations, Lecture 3.7: Cauchy-Euler equations
Differential Equations, Lecture 3.7: Cauchy-Euler equations.
In this lecture we encounter for the first time 2nd order ODEs with variable coefficients. The simplest of these are called "Cauchy-Euler equations", which have the form ax^2y''+bxy'+cy=0. The solutions of these ODEs are y(x)=x^r for some constant r, which is the root of a degree-2 polynomial. As before, there are three cases to be treated separately: distinct real roots, complex roots, and repeated roots. We also see how to convert Cauchy-Euler equations into ODEs with constant coefficients using a clever substitution.
Course webpage (with lecture notes, worksheets, etc.): http://www.math.clemson.edu/~macaule/math2080-online.html
Tracks in this Playlist
- Differential Equations, Lecture 1.1: What is a Differential Equation?
- Differential Equations, Lecture 1.2: Plotting solutions to differential equations
- Differential Equations, Lecture 1.3: Euler's method
- Differential Equations, Lecture 2.1: Separation of variables
- Differential Equations, Lecture 2.2: Initial value problems
- Differential Equations, Lecture 2.3: Newton's 2nd law of motion
- Differential Equations, Lecture 2.4: Solving 1st order inhomogeneous ODEs
- Differential Equations, Lecture 2.5: Linear differential equations
- Differential Equations, Lecture 2.6: Basic mixing problems
- Differential Equations, Lecture 2.7: Advanced mixing problems
- Differential Equations, Lecture 2.8: The logistic equation
- Differential Equations, Lecture 3.1: Second order ODEs
- Differential Equations, Lecture 3.2: 2nd order ODEs with constant coefficients
- Differential Equations, Lecture 3.3: The method of undetermined coefficients
- Differential Equations, Lecture 3.4: Simple harmonic motion
- Differential Equations, Lecture 3.5: Damped & driven harmonic motion
- Differential Equations, Lecture 3.6: Variation of parameters
- Differential Equations, Lecture 3.7: Cauchy-Euler equations
- Differential Equations, Lecture 3.8: Power series solutions
- Differential Equations, Lecture 3.9: The method of Frobenius
- Differential Equations, Lecture 4.1: Basic matrix algebra
- Differential Equations, Lecture 4.2: Eigenvalues and eigenvectors
- Differential Equations, Lecture 4.3: Mixing problems with two tanks
- Differential Equations, Lecture 4.4: Solving 2x2 systems of ODEs
- Differential Equations, Lecture 4.5: Phase portraits with real eigenvalues
- Differential Equations, Lecture 4.6: Phase portraits with complex eigenvalues
- Differential Equations, Lecture 4.7: Phase portraits with repeated eigenvalues
- Differential Equations, Lecture 4.8: Stability of phase portraits
- Differential Equations, Lecture 4.9: Variation of parameters for systems
- Differential Equations, Lecture 5.1: What is a Laplace transform?
- Differential Equations, Lecture 5.2: Properties & applications of the Laplace transform
- Differential Equations, Lecture 5.3: Laplace transforms of piecewise functions
- Differential Equations, Lecture 5.4: Laplace transforms of periodic functions
- Differential Equations, Lecture 5.5: Impulse functions
- Differential Equations, Lecture 5.6: Convolution
- Differential Equations, Lecture 6.1: Introduction to Fourier series
- Differential Equations, Lecture 6.2: Computing Fourier series
- Differential Equations, Lecture 6.3: Fourier sine & cosine series
- Differential Equations, Lecture 6.4: Complex Fourier series
- Differential Equations, Lecture 6.5: Applications of Fourier series
- Differential Equations, Lecture 6.6: Boundary value problems
- Differential Equations, Lecture 7.1: The heat equation
- Differential Equations, Lecture 7.2: Different boundary conditions.
- Differential Equations, Lecture 7.3: The transport equation
- Differential Equations, Lecture 7.4: The wave equation
- Differential Equations, Lecture 7.5: Harmonic functions
- Differential Equations, Lecture 7.6: Laplace's equation
- Differential Equations, Lecture 7.7: The 2D heat equation
- Differential Equations, Lecture 7.8: The 2D wave equation
- Differential Equations, Lecture 8.1: Modeling with nonlinear systems
- Differential Equations, Lecture 8.2: Linearization and steady-state analysis
- Differential Equations, Lecture 8.3: Predator-prey models