Math 22B: Differential Equations
Spring Quarter, 2008
Section 2
Announcement
Final solutions are posted below. Course grades will be posted by next Friday.
Instructor
John Hunter
e-mail: jkhunter@ucdavis.edu
Phone: (530) 752-3189
Office: 3230 MSB
Office hours: MWF 1:00–2:00 p.m.
Lectures: MWF 2:10–3:00 p.m., 2205 Haring Hall
Important dates:
- Last day to add: Tuesday, April 15, 2008
- Last day to drop: Friday, April 25, 2008
- Last class: Wednesday, June 4, 2008
- Academic holiday: Monday, May 26
TAs
- Martha Schott (3229 MSB)
Office Hours: W 9:45– 11:00 a.m.
- Josh Oyoung (2232 MSB)
Office Hours: T 2:00– 3:15 p.m., Th 3:00– 4:15 p.m.
Exams
There will be two in-class midterms and a final.
There will be no makeup exams.
Midterm 1
Midterm 1: Wednesday, April 23, 2:10–3:00 p.m.
Midterm 1 will cover the following topics.
- Sec. 1.1: Mathematical models leading to ODEs.
- Sec. 1.2: Some simple ODEs.
- Sec. 1.3: Classification of ODEs.
- Sec. 2.1: Integrating factor method for linear ODEs.
- Sec. 2.2: Separation of variables.
- Sec. 2.4: Existence/uniqueness theorems. Superposition principle for linear ODEs.
- Sec. 2.5: Autonomous ODEs. Phase lines, equilibria, and stability.
Midterm 1: Sample exam
Solutions to sample midterm questions
Midterm 1: Solutions
Midterm 2
Midterm 2: Wednesday, May 21, 2:10–3:00 p.m.
Midterm 2 will cover Sec. 3.1–3.6 and 3.8 on second order linear
ODEs:
- Sec. 3.1: Constant coefficient homogeneous ODEs
- exponential solutions
- characteristic equation
- solution of homogeneous ODEs when characteristic equation has distinct real roots
- Sec. 3.2: Fundamental solution sets
- superposition principle for homogeneous equations
- fundamental solution sets
- Wronskians
- solution of initial value problems
- Sec. 3.3: Linear independence
- linear dependence and independence of functions
- relationship between linear independence and the Wronskian
- Abel's theorem
- Sec. 3.4: Complex roots of characteristic equation
- Complex numbers
- Euler's formula and complex exponentials
- solution of homogeneous ODEs when characteristic equation has complex roots
- Sec. 3.5: Reduction of order
- solution of homogeneous ODEs when characteristic equation has repeated root
- Sec. 3.6: Nonhomogeneous ODEs and method of undetermined coefficients
- superposition principle for nonhomogeneous ODEs
- expression for general solution of nonhomogeneous ODE
in terms of particular solution and solutions of homogeneous ODE
- use of method of undetermined coefficients to find particular solutions
- Sec. 3.8: Unforced vibrations
- natural frequency of an undamped vibration
- effects of small and large damping (underdamped and overdamped vibrations)
- Sec. 3.9: Forced vibrations
Midterm 2: Sample questions 1 (Questions 4, 6 are not representative of the material we've covered this quarter)
Midterm 2: Sample questions 2 (Question 5 is not representative)
Solutions: Sample midterm questions 1
Solutions: Sample midterm questions 2
Solutions: Midterm 2
Final
Final (Exam Code B): Saturday, June 7, 10:30 a.m. – 12:30 p.m.
The final exam will be comprehensive. Topics covered will be those
listed above for the midterms plus material on linear systems:
- Sec. 7.1: First order linear systems
- Sec. 7.2–7.3: Linear algebra
- Sec. 7.4: Basic theory of linear systems
- Sec. 7.5: Homogeneous constant coefficient linear systems with real eigenvalues
- saddlepoints
- nodes (stable and unstable)
- Sec. 7.6: Homogeneous constant coefficient linear systems with complex eigenvalues
- spiral points (stable and unstable)
- centers
- Sec. 7.7: Fundamental matrices and the matrix exponential.
These topics will be emphasised a little on the final in comparison with the earlier topics.
Final: Sample questions 1
Solutions: Sample questions 1
Final: Sample questions 2 (Ignore Question 10 on variation of parameters, which we didn't cover
this quarter)
Solutions: Sample questions 2 (Unfortunately, the solutions I have don't exactly match the questions)
Solutions: Final Exam
Homework
Homework problems will be assigned on this web page, but homework will not be collected or graded. Selected homework solutions will be posted
after the suggested completion date.
Course Grade
The course grade will be based on (weights in parentheses):
- Midterms (30% each)
- Final (40%)
Text for Math 22B
Elementary Differential Equations and Boundary Value Problems, William E. Boyce and Richard C. DiPrima,
Eighth edition, John Wiley & Sons, 2005.
The course will cover Chapters: 1, 2, 3, 6, 7
- First order differential equations
- Second order linear equations
- Laplace transforms
- First order linear systems
The department syllabus
gives a detailed outline.
The publisher has a companion
web site for the text, which includes Maple, MATLAB and Mathematica files
for ODEs, and information about the ODE architect software included with the
text.
An additional resource for Math 22B is Craig Tracy's
Lectures on Ordinary Differential Equations.
Computer Accounts
The class will not have assignments that involve computation.
However, if you want to use numerical packages such as MATLAB to increase your understanding
of differential equations, you can sign up
with the mathematics department for a class account.
I will post some simple MATLAB files for ODEs here,
or you can write your own.
Homework Assignments
Set 1 (Friday, April 4)
- Sec. 1.1, p.7: 22, 23
- Sec. 1.2, p.15: 3, 8, 15, 17
- Sec. 1.3, p.24: 1– 6, 9, 15, 16, 29, 30
Solutions
Figures for solutions
Set 2 (Friday April 11):
- Sec 2.1, p. 39– 41: 14, 16, 20, 22 (b) (c), 30
- Sec 2.2, p. 47– 50: 1, 3, 7, 13, 26, 30 (a)--(e)
- Sec 2.4, p. 75–77: 1, 6, 7, 9, 13, 14, 21, 23, 26, 27, 28, 32
Solutions
Set 3 (Friday, April 18):
- Sec. 2.5, p. 88–94: 2, 3, 4, 5, 9, 13, 20, 25, 27, 28
- Sec. 2.7, p. 108–109: 1, 4 [only do these if you can use MATLAB], 20, 21
Read Section 2.7 on Numerical approximations on your own. We won't cover it in class lectures.
Solutions
Figure 1 for solutions
Figure 2 for solutions
Set 4 (Friday, May 2):
- Sec. 3.1, p. 142–143: 2, 5, 6, 9, 12, 16, 21, 27, 28
- Sec. 3.2, p. 151–153: 1, 2, 4, 5, 8, 12, 13, 14, 15, 16, 21, 24, 26, 27
Solutions
Set 5 (Friday, May 9):
- Sec. 3.3, p. 158–159: 1, 2, 5, 6, 9, 12, 17, 21, 27, 28
- Sec. 3.4, p. 164–166: 2, 4, 5, 11, 12, 17, 19, 27, 31, 38, 39
- Sec. 3.5, p. 172–174: 1, 4, 7, 12, 18, 20, 21, 22
Solutions
Set 6 (Friday, May 16):
- Sec. 3.6, p. 184–186: 1, 2, 5, 9, 10, 11, 13, 15, 27, 29
- Sec. 3.8, p. 203–206: 2, 16, 19, 24, 27, 30
Solutions
Set 7 (Friday, May 23):
- Sec. 3.9, p. 214–216: 17, 18 [rough, qualitative plots are fine in 17 (c) and 18 (b)]
- Sec. 7.2, p. 372–374: 1, 3, 4, 20, 22
- Read Sec. 7.2–7.3 in the text and review your linear algebra (especially eigenvalues
and eigenvectors)
Solutions
Figures
Set 8 (due Wednesday, June 4):
Sec. 7.5, p. 398–401: 2 (don't plot direction field), 11, 15, 24, 25, 27
- Sec. 7.6, p. 410–413: 2 (don't plot direction field), 7, 9, 13, 28
- Sec. 7.7, p. 420–422: 3, 11, 14, 17, 18
- Sec. 7.9 p. 439–440: 1, 3, 11
Solutions
Figures
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