m23f05 at math.dartmouth.edu(email to
ahbwill be ignored)
Water waves are described by PDEs
ifferential equation relate variables and their (possibly higher or partial)
These are universal in the sciences, with applications ranging from mechanics
through quantum physics, ecology, biology, engineering, finance and
statistics, to name a few fields. We will learn solution techniques,
building on calculus from Math 3, 8, and 13, such as separation
of variables, constant-coefficient methods, application of linear
algebra to systems of equations, and Fourier series and transforms.
One goal of the course will be to apply these methods to differential
equations arising in modeling the natural world.
We will culminate with partial differential equations such as Laplace's
equation, the wave and heat equations, and eigenvalue problems.
These days, computers enable accurate solution and visualization of differential equations, and this course has a key computer component (using Matlab, or any other equivalent package that you may prefer). As well as improving your understanding, this will teach you valuable tools that have become the bread-and-butter of science and engineering.
Lectures: MWF 10am, X-hour is Thurs 12 noon, location Bradley 105. I will announce X-hour usage each week; it will vary from displaced lectures to problem-solving to review, or empty. 4-5 quizzes will be given in lecture throughout the term. Office hours 4-5pm Mon, 3-5pm Tues.
|week||date||reading||homework (due following Wed) / daily or weekly topics / info|
|1||Sep 21 W||website, 1.1-1.3||HW1 assigned. Direction fields, introduction to initial-value problems, starting with computing.|
|22 Th X-hour||Overview of Matlab, important to get it set up and check simple commands work|
|23 F||2.1, 2.2||First-order linear ODEs, separation of variables|
|26 M||2.4, 2.6||Existence and uniqueness, exact equations.|
|2||28 W||2.7||HW2 assigned (get |
|29 Th X-hour||(free)|
|30 F||2.3, 2.5||Modeling and population models.|
|Oct 3 M||3.1||Second-order linear homogeneous ODEs: characteristic equation|
|3||5 W||3.2||revised HW3 assigned. Quiz 1 (on Ch.2, no Matlab; solutions). Fundamental solutions, Wronskian.|
|6 Th X-hour||this||(read crash course on 2-by-2 matrices in your own time)|
|7 F||3.3||Wronskian, linear independence|
|10 M||3.4||Complex roots|
|4||12 W||3.5, 3.6||HW4 assigned (get resonance.m). Repeated roots, method of undetermined coefficients.|
|13 Th X-hour||3.7||catch-up lecture on 2nd-order ODEs: variation of parameters|
|14 F||3.8||Mechanical, electrical vibrations. Links: unforced, Q-factor, forcing 1, 2, Tacoma videos 1, 2, 3.|
|17 M||3.9||Driven oscillators and resonance.|
|5||19 W||5.1, 5.2||HW5 assigned. Power series, series solutions|
|20 Th X-hour||Quiz 2 (on Ch.3, no Matlab; solutions), and power series problems. Matlab Taylor convergence demo (calls show_zser).|
|21 F||5.3, 5.4||series solutions about an ordinary point, Euler equations. power series applet.|
|24 M||7.1, 7.2||Systems of first-order equations, existence and uniqueness review, review row reduction.|
|6||Midterm: Wednesday October 26th
(on everything so far).|
Also an open-book take-home part (solutions, codes:
|28 F||7.3||Take-home exam due. Linear algebra review. Applet to visualize Ax.|
|30 M||7.3||Eigenvalues, eigenvectors.|
|7||Nov 2 W||7.4, 7.5||HW6 assigned. Homogeneous systems. Applets: click pplane, and local|
|3 Th X-hour||7.6||Handling unusual eigenvalues.|
|4 Fr||(no lecture: Alex out of town)|
|7 M||7.8, 9.1||Phase portraits of linear systems.|
|8||9 W||9.2, 9.3||HW7 assigned. Almost-linear systems, pendulum.|
|10 Th X-hour||Quiz 3 (on Ch. 7, excluding 7.7 and 7.9; solutions).|
|11 Fr||10.1||Boundary-value problems|
|14 M||10.2, 10.3||Fourier series, convergence.|
|9||16 W||10.4||HW8 assigned.|
|17 Th X-hour||(Fourier review?)|
|18 Fr||10.5||Separation of variables. PDEs: heat equation, applet, and.|
|21 M||10.6, 10.7||HW9 assigned. Heat w/ mixed boundary conditions. Wave equation, applet (maximize # masses), odd modes and pulse.|
|........... 22 Tu - 27 Su Thanksgiving recess .............|
|10||28 M||10.8||Laplace's equation.|
|30 W||Quiz 4 (on Ch. 9.1-3 and 10.1-6; solutions). HW9 due. In-class worksheet. (no lecture: Alex out of town W and Th)|
|Dec 2 Fr||Review session (during usual lecture time). Recurring themes and Review choosing methods.|
|Final Exam: Monday December 5th at 8:00 am - 11:00 am, Bradley 102. Single sheet of notes allowed, no algebraic/graphing calculators. Note this exam will not be given early to accommodate travel plans. (solutions).|
Grade weighting: Your overall grade will be computed according to HW 20%, Quizzes 15%, Midterm 25%, Final 40%. Note that although HW has a low weighting, it is the main chance you get to practise the material and get feedback.
Honor principle. Exams: no help given or received. Homework: no copying, however collaboration on approaches to problems is encouraged. Write-ups must be done individually.
Homework: I strongly encourage you to attempt the relevant homework problems before the next lecture. Leaving it all for Tuesday night is bad time management and risks you getting left behind in this fast-paced course. Please make your working/reasoning as clear as you can, write clearly, don't be scared of using lots of space on the page, and staple your work. Late homework will not be accepted (unless by prior arrangement for a valid reason). Your lowest HW score will be dropped.
Special needs: Let me know asap, certainly in first 2 weeks. Also stop by the Academic Skills Center in 301 Collis to register for support services.
Private tutoring: Tutor Clearinghouse may have private one-on-one tutors available for Math 23. The tutors are recruited on the basis that they have done well in the subject, and are trained by the Academic Skills Center. If a student receives financial aid, the College will pay for three hours of tutoring per week. If you would like to have a tutor, please go to 301 Collis and fill out an application as early in the term as possible.
dfieldto plot direction fields overlayed with ODE solutions: Download Matlab version (save to desktop). Similarly here is pplane7.m (save to desktop). Or you can use java applet versions from your browser.