This course covers techniques for solving a variety of classes of ordinary differential equations. We will primarily focus on methods for finding explicit solutions, rather than approximate numerical solutions. Along the way we will develop lots of techniques, some of them "tricks" but many of them broadly applicable in mathematics.
We will assume familiarity with calculus, especially the ability to compute standard integrals and derivatives of functions involving trigonometric functions, exponential functions, etc. We also assume basic familiarity with complex numbers. An understanding of basic linear algebra (matrix multiplication, linear transformations, determinants, eigenvectors) will be helpful, although we will give a crash review of the material as needed.
Some of the topics we will cover over the course of the semester include:
Elementary Differential Equations and Boundary Value Problems, 10th Edition - Boyce and DiPrima.
The material of this course will roughly follow chapters 1,2,3,4,5,7 of the textbook by Boyce and DiPrima. The lectures will include some of the material from these chapters, as well as some additional material and examples. We highly recommend following reading the corresponding sections of the textbook concurrently.
In addition to the lectures, there will be about twelve weekly problem sets. These will be listed on this website and should be handed to the labeled box on the 4th floor of the mathematics building. Graded problem sets can be found in the box on the sixth floor. Late homeworks will not be accepted.
It is very important to do all of the problem sets to the best of your ability, as this is the most effective way to absorb the material. While you are welcome to collaborate with your peers, you must attempt all problems on your own and your submitted solutions must be written out individually. Submissions which are copied or suspiciously similar may be rejected.
There will be two midterm exams and one final exam. The exams will generally follow the material from the problem sets but may include some additional conceptual problems to test your understanding. Each midterm will cover roughly a third of the course material. The final exam will cover all the material from the course, with slightly more emphasis on the content covered after the second midterm. Keep in mind that the first midterm is already in week 5 - don't be caught off-guard! The date of the final is determined by the university registrar.
Homeworks: 10%, midterm exams: 50%, final exam: 40%. We will drop the lowest homework grade.
UPDATE: all office hours will now be via Zoom. Please see https://www.wejoinin.com/sheets/kvmly for times, Zoom IDs, and the signup sheet.
| Date | Material | References | Problem Set |
|---|---|---|---|
| Week 1: Tuesday 1/21/20 and Thursday 1/23/20 | Introduction. Definition of ODE, linear versus nonlinear. Separable ODEs. | §1.1,§1.2,§1.3,§1.4,§2.2 |
Problem set 1
Problem set 1
(due Friday 1/31/20 by 5pm): §1.2: 7,8, §1.3: 1,2,3,4,5,6,14,20, §2.2: 1,2,3,5,7,8.
Solutions
|
| Week 2: Tuesday 1/28/20 and Thursday 1/30/20 | First order linear ODEs, solution by integrating factors. Exact ODEs. | §2.1,§2.6 |
Problem set 2
Problem set 2
(due Friday 2/7/20 by 5pm): §2.1: 2,6,9,13,30,35 §2.3: 8, §2.6: 1,4,9,15,25.
Note: for slope field plots, you're welcome to do them with a computer package such at Matlab or Mathematica, or any of the free utilities available online. However, you should also become comfortable doing them by hand.
Solutions
|
| Week 3: Tuesday 2/4/20 and Thursday 2/6/20 | Autonomous ODEs, equilibria, and graphical methods. Discussion of existence and uniqueness theorem. Euler's method. | §2.4,§2.5,§2.7,§2.8 |
Problem set 3
Problem set 3
(due Friday 2/14/20 by 5pm): §2.1: 33,38, §2.5: 2,11,15,23, §2.6: 24,31, §2.7 4,11.
Solutions
|
| Week 4: Tuesday 2/11/20 and Thursday 2/13/20 | Second order homogeneous ODEs with constant coefficients. The Wronksian. Complex roots. | §3.1,§3.2,§3.3 | Sample midterm. With solutions. |
| Week 5: Tuesday 2/18/20 and Thursday 2/20/20 | Repeated roots and reduction of order. Midterm 1 on 2/20 during class. Midterm 1 solutions | §3.4 |
Problem set 4
Problem set 4
(due Friday 2/28/20 by 5pm): §3.1: 10,12,16, §3.2: 5,9,14,15,16,17, §3.3: 6,11,17,19,21, §3.4: 3,11,14.
Solutions
|
| Week 6: Tuesday 2/25/20 and Thursday 2/27/20 | Inhomogeneous ODEs with constant coefficients. The method of reduction of undetermined coefficients. Variation of parameters. | §3.5,§3.6 |
Problem set 5
Problem set 5
(due Friday 3/6/20 by 5pm): §3.2: 29, §3.4: 23,31,37, §3.5: 5,9,10,20,34, §3.6: 3,5,6,17.
Solutions
|
| Week 7: Tuesday 3/3/20 and Thursday 3/5/20 | Complex roots of unity. Higher order ODEs with constant coefficients. Undetermined coefficients for higher order. | §4.1,§4.2,§4.3, handout |
Problem set 6
Problem set 6
(due Friday 3/20/20 by 11:59pm): §4.1: 9,10,15, §4.2: 1,2,5,8,10,11,14,15,21, §4.3: 1,2,6. Note: this problem set and all future ones are to be submitted online via CourseWorks.
Solutions
|
| Week 8: Tuesday 3/10/20 and Thursday 3/12/20 | Review of power series. Ordinary versus singular points. Series solution near ordinary point. | §5.1,§5.2 |
Problem set 7
Problem set 7
(due Monday 3/30/20 by 11:59pm): §5.1: 1,4,7,13,16,17,20,25, §5.2: 1,5,9.
Solutions
|
| Spring break: Tuesday 3/17/20 and Thursday 3/19/20 | No class | ||
| Week 9: Tuesday 3/24/20 and Thursday 3/26/20 | UPDATE: Spring break extended to 3/25/20. More on ordinary versus singular points. Series solution near ordinary point continued. Airy's equation. | §5.1,§5.2 | |
| Week 10: Tuesday 3/31/20 and Thursday 4/2/20 | More on series solutions near ordinary points. Euler equations. Regular versus irregular singular points. Series solution near regular singular point. | §5.2,§5.3 |
Problem set 8
Problem set 8
(due Monday 4/13/20 by 11:59pm): §5.2: 21, §5.3: 2,7,11,13, §5.4: 1,2,8,9,19,27,28,31,37.
Solutions
|
| Week 11: Tuesday 4/7/20 and Thursday 4/9/20 | More on Euler's equations. Midterm 2 available 4/9 - 4/12 (timed online test, see Canvas announcements for details). Note: no class on 4/9. Midterm 2 solutions | §5.2,§5.3,§5.4 | Sample midterm. With solutions. |
| Week 12: Tuesday 4/14/20 and Thursday 4/16/20 | Series solutions near regular singular points. Crash course in linear algebra. Systems of algebraic linear equations. Systems of first order linear ODEs. | §5.5,§5.6,§7.1,§7.2,§7.3 |
Problem set 9
Problem set 9
(due Monday 4/20/20 by 11:59pm): §5.5: 1,5,10,12,13, §5.6: 5,9
Solutions
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| Week 13: Tuesday 4/21/20 and Thursday 4/23/20 | More on systems of first order linear ODEs. Complex eigenvalues. | §7.4,§7.5,§7.6. |
Problem set 10
Problem set 10
(due Monday 4/27/20 by 11:59pm): §7.1: 4,5,12,15, §7.2: [1],[8],12,14,[21],23, §7.3: [1],[2],6,25. The problems in square brackets are purely optional (not extra credit), intended to improve your comfort level with relevent linear algebra.
Solutions
|
| Week 14: Tuesday 4/28/20 and Thursday 4/30/20 | Repeated eigenvalues. Fundamental matrices. Additional topics. | §7.7,§7.8. |
Problem set 11
Problem set 11
(due Wednesday 5/6/20 by 11:59pm): §7.5: 1,11,12,16, §7.6: 1,9, §7.8: 1,5.
Solutions
|
| Final exam | The projected date for the final exam is Tuesday 5/12/20. UPDATE: details to follow. Sample final (note: this is not necessarily indicative of the format of our final) | Bonus problem set (due Friday 5/15/20 by 11:59pm). |