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)
Differential Equations, Lecture 3.9: The method of Frobenius. In the previous lecture, we saw how many 2nd ODEs have solutions that are power series. In this lecture, we learn when this is the case, which is due to a theorem of Frobenius. In particular, if the center x_0 of the power series is a so-called "ordinary point", then the ODE will have a power series solution. Otherwise, x_0 is said to be a "singular point," and there are two types of these. If x_0 is an "regular singular point", then there is a generalized power series solution, which is simply a power series times x^r, where r could be a negative number, fraction, or complex number. In this lecture, we learn what these mean, we discuss the radii of convergence of these solutions, and we conclude with an example of an ODE that has as generalized power series solution. Course webpage (with lecture notes, worksheets, etc.): http://www.math.clemson.edu/~macaule/math2080-online.html