MATH
245 Mathematics of Physics and Engineering
Syllabus
[pdf]
Lectures 
MWF
10:0010:50 am in MHP
105 
Instructor 

Office 

Email 
kzuev@usc.edu (please
include “245” in the subject line) 
Phone 
(213) 7402393 
Office
Hours 
MF 1:302:30
pm, W 2:303:30, or by appointment 
Discussions 
TTh
8:008:50 am, 9:009:50 am, 10:0010:50 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: firstorder
differential equations; secondorder 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 closedbook 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 closedbook.
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, 810am, location
to be announced. The final exam is closedbook 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, 810am 
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]
