Bernd Schroeder

Differential Equations

 

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The videos below are part of my remotely delivered differential equations course. Video presentations are seamlessly integrated with my differential equations book, and the numbering of the modules is the same as in the book. Nonetheless, these presentations can be used as stand-alones without my book or for self-study. The idea was and is to teach mathematics, no more and no less. Presentations are sorted by topic. The names of the videos should be self-explanatory. For the "license", click here.

Note that the videos were shot almost without re-takes in 4-8 hour sessions. Thus some of them will have small "hiccups" and some may need to be re-shot. On the other hand, I do like the rather natural "classroom feel" of the videos. No classroom presentation is ever flawless, and it can be fun to catch the teacher misspeaking. (I kept an extra presentation on Laplace transforms for integral equations for exactly that reason.)

The videos were produced with tegrity. For best performance, they should be downloaded and then run with Internet Explorer as the default browser. The slides were produced with the LaTeX beamer package.

Introduction.

This is an overview of the course and of my philosophy for the course, the text and the videos. There is some popping on the sound. I may need to re-shoot this one. 

Video. (26:13min, 81MB)                Slides.       

      

Module 1. Modeling - some important examples.

  • Derivation of the differential equation for a spring-mass-system. (Could be viewed right before Module 3.)
    Video.   (9:49min, 29MB)               Slides.
  • Derivation of the differential equation for an LRC circuit. (Could be viewed right before Module 3.)
    Video.   (7:11min, 21MB)               Slides.
  • Derivation of the differential equations for a multi-loop circuit, Kirchhoff's laws. (Could be viewed right before Module 6.)
    Video.   (7:27min, 23MB)               Slides.
  • Derivation of the equation for an oscillating string. (Could be viewed right before Module 7.)
    Video.  (16:23min, 49MB)               Slides.  
  • Derivation of the heat equation. (Could be viewed right before Module 7.)
    Video.  (10:08min, 32MB)               Slides.

 

Module 2. First order differential equations.

  1. General solution of separable differential equations.
    Video.   (8:47min, 27MB)                Slides.
    Solution of an initial value problem for a separable differential equation. (Reviews integration by substitution and integration by parts.)
    Video.  (12:18min, 36MB)                Slides.
    Setting up mixing problems and solving them with separable differential equations.
    Video.  (15:41min, 48MB)                Slides.
  2. Linear first order differential equations.
    Video.   (8:59min, 28MB)                Slides.
  3. Bernoulli equations.
    Video.  (15:36min, 47MB)                Slides.
  4. Homogeneous first order equations.
    Video.   (9:01min, 27MB)                Slides.
  5. Exact differential equations.
    Video.   (8:27min, 26MB)                Slides.
  6. How to recognize types of first order equations and how to review them. (I misspeak twice, possible candidate for re-shoot.) (Section 2.7 in the book.)
    Video.  (15:58min, 47MB)                Slides. 

Module 3. Second order constant coefficient differential equations.

  1. Solution of linear homogeneous second order differential equations with constant coefficients. (Sections 3.1-3.3 in the book.)
    Video.  (18:54min, 55MB)                Slides.
  2. The method of undetermined coefficients. (Section 3.4 in the book.)
    Video.  (21:52min, 65MB)                Slides.
  3. The method of undetermined coefficients when the forcing function solves the homogeneous equation. (Section 3.4 in the book.) There is some popping on the sound. I may need to re-shoot this one.
    Video.  (21:26min, 66MB)                Slides. 
  4. The formula for Variation of Parameters. (Section 3.5 in the book.) 
    Video.  (19:49min, 62MB)                Slides. 
  5. The formula for Variation of Parameters. (Solves some rather nasty integrals. Typo on one slide, candidate to be re-shot.)
    Video.  (19:24min, 61MB)                Slides. 
  6. Cauchy-Euler equations. (Even though these equations do not have constant coefficients, they fit in quite well at this spot.)
    Video.  (14:09min, 45MB)                Slides.

 

Module 4. Qualitative and Numerical Approaches.

  1. Direction fields.
    Video.    (6:59min, 23MB)               Slides.
    Autonomous differential equations.
    Video.   (11:19min, 36MB)               Slides.
  2. Euler's method.
    Video.   (21:06min, 75MB)               Slides.
  3. Improved Euler method and Runge-Kutta methods.
    Video.   (26:33min, 90MB)               Slides.
  4. Finite difference method.
    Video.   (25:42, 77MB)                    Slides.

 

Module 5. Theory of linear differential equations. (One long video only.)

  • There is some popping on the sound. I may need to re-shoot this one.
    Video.  (39:22min, 122MB)               Slides. 

 

Module 6. Laplace transforms.

  1. Introduction to Laplace transforms.
    Video.   (23:38min, 71MB)               Slides. 
  2. Solving initial value problems with Laplace transforms.
    Video.   (13:46min, 43MB)               Slides.
  3. Solving systems of differential equations with Laplace transforms.
    Video.   (12:20min, 39MB)               Slides. 
  4. An initial value problem that involves damped trigonometric functions.
    Video.   (20:29min, 65MB)               Slides.
  5. Step functions and Laplace transforms.
    Video.   (19:55min, 61MB)               Slides.
    Delta functions and Laplace transforms. 
    Video.   (20:05min, 63MB)               Slides.           Animation.
  6. Laplace transforms and convolutions.
    Video.   (14:44min, 47MB)               Slides. 
    Laplace transforms of periodic functions.
    Video.   (23:46min, 72MB)               Slides. 
    Laplace transforms of integral equations. 
    Video.   (10:33min, 34MB)               Slides.
    The first video (12:07min, 38MB) I made for this topic approaches the partial fractions in a clumsy manner. It has a heartfelt comment about my own abilities slightly after the 5:00 minute mark and another scary moment after the 8:45 mark. (Unlike the other jokes, neither one was planned.)

 

Module 7. Separation of variables.

  • Solving the equation for the oscillating string. (Sections 7.1-7.3 in the book.)
    Video.   (32:46min, 102MB)             Slides.          Animation 1.          Animation 2.         Animation 3.
  • Deriving the Legendre equation. (Section 7.4 in the book.) This presentation plus the presentation on Legendre polynomials (see under series solutions below) provide most of the mathematics for the quantum mechanical description of the hydrogen atom.
    Video.     (22:00min, 71MB)             Slides.  
  • Deriving the Bessel equation. (Section 7.4 in the book.)
    Video.     (12:39min, 41MB)             Slides. 
  • Eigenvalues of the Laplace operator.
    Video.      (9:32min, 31MB)              Slides. 

Module 8. Series solutions.

Review

  • Taylor polynomials (summary).
    Video.   (13:45min, 43MB)              Slides. 
  • Power series (summary).
    Video.   (13:14min, 43MB)              Slides. 
  1. Series solutions about ordinary points.
    Video.   (13:42min, 43MB)              Slides.
    Radius of convergence of power series solutions.
    Video.    (11:44min, 37MB)             Slides. 
  2. Legendre polynomials. This presentation plus the presentation on deriving the Legendre equation (see separation of variables above) provide most of the mathematics for the quantum mechanical description of the hydrogen atom.
    Video.  (31:47min, 101MB)              Slides.
  3. Method of Frobenius.
    Video.    (19:14, 60MB)                  Slides.
    Method of Frobenius: An example in which we only get one solution. (Also an example of a specific Bessel equation.) There is some popping on the sound. I may need to re-shoot this one.
    Video.    (26:05min, 75MB)             Slides. 
  4. Bessel equations.
    Video.    (25:57min, 82MB)             Slides.           Animation 1.              Animation 2.
  5. Reduction of order. This topic fits here, because the fact that the method of Frobenius sometimes gives only one solution motivates reduction of order. There is some popping on the sound. I may need to re-shoot this one.
    Video.    (12:20min, 37MB)             Slides.  

 

Module 9. Systems of linear differential equations.

  1. Translation from higher order to first order systems.
    Video.     (10:19min, 34MB)            Slides.
  2. Matrix multiplication.
    Video.      (9:00min, 28MB)            Slides.
  3. Diagonalizable systems of linear differential equations with constant coefficients.
    Video.    (36:02min, 112MB)            Slides.
    Diagonalizable systems of linear differential equations with constant coefficients, complex eigenvalues.
    Video.      (16:36min, 56MB)           Slides.
  4. Non-diagonalizable systems of linear differential equations with constant coefficients.
    Video.      (13:28min, 44MB)           Slides.

 

 

 

"License".

Obviously, if I post something, I want people to use it. So do it. If you like the videos, try the book. If you are a teacher, feel free to use the videos and the slides in classes. The goal is to get people to do better in mathematics.

One caveat: If you want to create an on-line course with the videos, note that I have already done so. Please consider sending your students to my course  :)

Sample course outline (Winter term 2008-2009, 3 semester credit hours)

Slides for a presentation on on-line delivery