|
MATH
245 Mathematics of Physics and Engineering
Syllabus
[pdf]
Lectures |
MWF
10:00-10:50 am in MHP
105 |
Instructor |
|
Office |
|
Email |
kzuev@usc.edu (please
include “245” in the subject line) |
Phone |
(213) 740-2393 |
| Office
Hours |
MF 1:30-2:30
pm, W 2:30-3:30, or by appointment |
| Discussions |
TTh
8:00-8:50 am, 9:00-9:50 am, 10:00-10:50 am in KAP
159 |
| Teaching
Assistant |
Andrew
Williams |
Coure
Description
Differential
equations are used in all fields of science and engineering. The
main goal of this course is to provide an introduction to some fundamental
properties of differential equations, and to present some of the
main methods for finding their solutions. Topics will include: first-order
differential equations; second-order linear differential equations;
determinants and matrices; systems of linear differential equations;
Laplace transforms. |
Prerequisites
Textbooks
J.R.
Brannan and W.E. Boyce, Differential Equations: An Introduction
to Modern Methods and Applications, 2nd ed., Wiley, 2011. |
Course
Plan
The
following is a tentative outline of the material to be covered
this term.
Sections |
Topic |
#
of Lectures |
§
1.1, 1.2, 1.4 |
Introduction |
2 |
| §
2.1 – 2.5 |
First
order differential equations |
6 |
| § 3.1
- 3.6 |
Systems
of two first order equations |
6 |
| § 4.1
– 4.7 |
Second
order linear equations |
8 |
| § 6.1
– 6.6 |
Systems
of first order linear equations |
7 |
| § 5.1
- 5.8 |
The
Laplace transform |
8 |
| § 7.1
– 7.3, 7.5 |
Nonlinear
Differential Equations |
4 |
|
Grading
Quizzes |
10% |
First
Midterm |
25% |
Second
Midterm |
25% |
Final |
40% |
|
Homework
Suggested
homework problems will be posted here
after each lecture. These problems will be assigned but not
collected for grading. Homework is considered to be a vital
part of the learning experience in the class, and is of crucial
importance to successful completion of the course. A respectable
performance on quizzes and exams can be realized by all students
if attention and energy are given to the timely completion
of assigned homework problems.
|
Quizzes
A
quiz will be given each week on Tuesday, except for the first
week of class, and for the two weeks when midterms are held.
The quiz problems will be similar to homework problems assigned
in the previous week. For example, the quiz problems on Jan
17 will be similar to homework problems assigned on Jan 9, 11,
and 13. The two lowest quiz grades will be dropped in the final
grade calculations. All quizzes are closed-book and no calculators
are allowed or needed. |
Midterm
Exams
There
will be two (one hour) midterm exams: Monday, February 13 (exam
1) and Monday, March 26 (exam 2). The 2nd exam will cover material
after the 1st exam. Both exams will be given in regular class
time. The place will be announced later. Both exams are closed-book.
No calculators are allowed or needed. |
Final
Exam
The
final exam is comprehensive and will be held at the time specified
in the
University Schedule of Classes: Monday, May 7, 8-10am, location
to be announced. The final exam is closed-book and no calculators
are allowed or needed. |
Important
Dates
Homework |
Weekly
on Mondays, Wednesdays, and Fridays |
Quizzes |
Jan
18, 24, 31, Feb 7, 21, 28, Mar 6, 20, Apr 3, 10, 17,
24 |
First
Midterm |
Monday,
February 13 |
Second
Midterm |
Monday,
March 26 |
Final |
Monday,
May 7, 8-10am |
Expectations
Official
announcements, homework assignments, and midterm solutions
will be posted on the course website. You are expected
to check the course website on a regular basis. You
are encouraged to read the appropriate sections of the
textbook in advance and discuss the homework assignments
with other students. |
Behavior
Behavior
that persistently or grossly interferes with classroom
activities is considered disruptive behavior and may
be subject to disciplinary action. Such behavior inhibits
other students’ ability to learn and an instructor’s
ability to teach. A student responsible for disruptive
behavior may be required to leave class pending discussion
and resolution of the problem and may be reported to
the Office
of Student Judicial Affairs for disciplinary action.
In particular, the use of cell phones during class or
conversation is disruptive behavior. |
Academic Integrity
All
students are responsible for maintaining standards of
academic integrity. In particular, collaboration, use
of notes, or any electronic devices during quizzes,
midterms or the final are strictly prohibited. |
Useful
Links
|
Schedule
Date |
Lecture |
Topic |
Homework |
January
9 |
|
Differential
Equations: An Introduction |
Sec.
1.1 #3,17,33,38 |
| January
11 |
|
Classification
of Differential Equations and Method of Integrating Factors |
Sec.
1.4 #5,6
Sec.
1.2 #17,30
|
| January
13 |
|
First
order ODEs: Separable Equations |
Sec.
2.1 #2,7,11(a),18(a) |
| January
16 |
N/A |
MLK
Day |
N/A |
| January
18 |
|
Modeling with First Order ODEs |
Sec.
2.2 #2,23 |
| January
20 |
|
Existence and Uniqueness of Solutions |
Sec.
2.3 #1,9,15 |
| January
23 |
|
Autonomous
and Exact Equations |
Sec.
2.5 #1(b),3(b),5(b),9(b) |
| January
25 |
|
Exact
Equations and Integrating Factors |
Sec.
2.5 #11(a), 13(solve the IVP), 26(a) |
| January
27 |
|
Systems
of Two Linear Algebraic Equations: A Review |
Sec.
3.1 #13,15,17,33 |
| January
30 |
|
Systems
of Two First Order ODEs |
Sec.
3.2 #5,10(a),15(a),26 |
| February
1 |
|
Homogeneous
Autonomous Systems: Real and Different Eigenvalues |
Sec.
3.3 #3,11,14,16 |
| February
3 |
|
Homogeneous
Autonomous Systems: Complex Eigenvalues |
Sec.
3.4 #3,5,7,9 |
| February
6 |
|
Matrices,
Determinants, and Complex Variables: a Brief Overview |
Sec.
A1: 1; A3: compute the determinant 6,7; B: 12,21,27 |
| February
8 |
|
Homogeneous
Autonomous Systems: Repeated Eigenvalues |
Sec.
3.5: find the general solution 3,5; solve the IVP 9,11 |
| February
10 |
|
Classification
of Phase Portraits |
Classify
phase portraits: Sec. 3.3 #10; Sec. 3.4 #6; Sec. 3.5 #4 |
| February
13 |
N/A |
Midterm
1 |
N/A |
| February
15 |
|
Second
order Linear ODEs:
Definitions and Examples |
Sec.
4.1 #1,2,3,4,5,18 |
| February
17 |
|
Theory
of Second Order Linear Homogeneous ODEs |
Sec.
4.2 #1,3,5 |
| February
20 |
N/A |
President's
Day |
N/A |
| February
22 |
|
Theory
of Second Order Linear Homogeneous ODEs - II |
Sec.
4.2 #11,15,17,23 |
| February
24 |
|
Linear
Homogeneous Second Order ODEs with Constant Coefficients |
Sec.
4.3 #9(a), 15(a), 17(a), IVP 37, 45, 47 |
| February
27 |
|
Nonhomogeneous
Equations: Method of Undetermined Coefficients |
Sec.
4.5 #1, 2, 13 |
| February
29 |
|
Method
of Undetermined Coefficients II |
Sec.
4.5 #3, 5, 15, 23(a) |
| March
2 |
|
Variation
of Parameters for Linear First Order Systems |
Sec.
4.7 #3, 5, 9 |
| March
5 |
|
Variation
of parameters for Linear Second Order ODEs |
Sec.
4.7 #11, 16, 23 |
| March
7 |
|
The
Laplace Transform |
Sec.
5.1 # 7, 10, 17, 25 |
| March
9 |
|
Properties
of the Laplace Transform |
Sec.
5.2 # 3, 5, 7, 9, 19 |
| March
12 |
N/A |
Spring
Recess |
N/A |
| March
14 |
N/A |
Spring
Recess |
N/A |
| March
16 |
N/A |
Spring
Recess |
N/A |
| March
19 |
|
The
Inverse Laplace Transform |
Sec.
5.3 # 9, 17, 23 |
| March
21 |
|
Solving
Initial Value Problems with Laplace Transforms |
Sec.
5.4 # 3, 11, 15 |
| March
23 |
|
Discontinuous
Functions and Periodic Functions |
Sec.
5.5 # 5, 9, 13, 23 |
| March
26 |
N/A |
Midterm
2 |
N/A |
| March
28 |
|
Impulse
Functions |
Sec.
5.7 # 1, 3, 5 |
| March
30 |
|
Convolution
Integrals and Their Applications |
Sec.
5.8 # 5, 9, 19 |
| April
2 |
|
Systems
of First Order Linear ODEs: Definitions and Examples |
Sec.
6.1 # 2, 5, 9 |
| April
4 |
|
Basic
Theory of First Order Linear Systems |
Sec.
6.2 # 5, 9, 11 |
| April
6 |
|
Homogeneous
Linear Systems with Constant Coefficients |
Sec.
6.3 # 3, 7, 9 |
| April
9 |
|
Homogeneous
Linear Systems with Constant Coefficients: Complex Eigenvalues |
Sec.
6.4 # 1, 5, 7 |
| April
11 |
|
Fundamental
Matrices and the Exponential of a Matrix - I |
Sec.
6.5 # 3, 5, 7 |
| April
13 |
|
Fundamental
Matrices and the Exponential of a Matrix - II |
Sec.
6.5 # 9, 11, 17 |
| April
16 |
|
Nonhomogeneous
Linear Systems |
Sec.
6.6 # 1, finish example on Slide 5. |
| April
18 |
|
Autonomous
Nonlinear Systems and Stability |
Sec.
7.1 # 3, 5, 7, 9, 11 |
| April
20 |
|
Almost
Linear Systems |
Sec.
7.2 # 5(a,b,c), 7(a,b,c), 13(a,b,c), 19(a,b,c) |
| April
23 |
|
Competing
Species |
Sec.
7.3 # 1, 3, 5 |
| April
25 |
|
Periodic
Solutions and Limiting Cycles |
Sec.
7.5 # 3, 5, 11 |
| April
27 |
|
Review
of the Course |
N/A |
Solutions
Feb
13: Midterm 1 [pdf],
average score = 62.3, median = 67.5
Mar 26: Midterm 2 [pdf],
average score = 79.5, median = 82
Fun
Photo [jpg]
|
