Math245

MATH 245 Mathematics of Physics and Engineering

 Syllabus Schedule Solutions Fun

Syllabus [pdf]

 Lectures MWF 10:00-10:50 am in MHP 105 Instructor Konstantin Zuev Office KAP 424A 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
 MATH 226 Calculus III

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

 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.

 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.

 Grades will be posted on: Blackboard

Schedule

 Date Lecture Topic Homework January 9 1 [pdf] Differential Equations: An Introduction Sec. 1.1 #3,17,33,38 January 11 2 [pdf] Classification of Differential Equations and Method of Integrating Factors Sec. 1.4 #5,6 Sec. 1.2 #17,30 January 13 3 [pdf] First order ODEs: Separable Equations Sec. 2.1 #2,7,11(a),18(a) January 16 N/A MLK Day N/A January 18 4 [pdf] Modeling with First Order ODEs Sec. 2.2 #2,23 January 20 5 [pdf] Existence and Uniqueness of Solutions Sec. 2.3 #1,9,15 January 23 6 [pdf] Autonomous and Exact Equations Sec. 2.5 #1(b),3(b),5(b),9(b) January 25 7 [pdf] Exact Equations and Integrating Factors Sec. 2.5 #11(a), 13(solve the IVP), 26(a) January 27 8 [pdf] Systems of Two Linear Algebraic Equations: A Review Sec. 3.1 #13,15,17,33 January 30 9 [pdf] Systems of Two First Order ODEs Sec. 3.2 #5,10(a),15(a),26 February 1 10 [pdf] Homogeneous Autonomous Systems: Real and Different Eigenvalues Sec. 3.3 #3,11,14,16 February 3 11 [pdf] Homogeneous Autonomous Systems: Complex Eigenvalues Sec. 3.4 #3,5,7,9 February 6 12 [pdf] Matrices, Determinants, and Complex Variables: a Brief Overview Sec. A1: 1; A3: compute the determinant 6,7; B: 12,21,27 February 8 13 [pdf] Homogeneous Autonomous Systems: Repeated Eigenvalues Sec. 3.5: find the general solution 3,5; solve the IVP 9,11 February 10 14 [pdf] 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 15 [pdf] Second order Linear ODEs: Definitions and Examples Sec. 4.1 #1,2,3,4,5,18 February 17 16 [pdf] Theory of Second Order Linear Homogeneous ODEs Sec. 4.2 #1,3,5 February 20 N/A President's Day N/A February 22 17 [pdf] Theory of Second Order Linear Homogeneous ODEs - II Sec. 4.2 #11,15,17,23 February 24 18 [pdf] Linear Homogeneous Second Order ODEs with Constant Coefficients Sec. 4.3 #9(a), 15(a), 17(a), IVP 37, 45, 47 February 27 19 [pdf] Nonhomogeneous Equations: Method of Undetermined Coefficients Sec. 4.5 #1, 2, 13 February 29 20 [pdf] Method of Undetermined Coefficients II Sec. 4.5 #3, 5, 15, 23(a) March 2 21 [pdf] Variation of Parameters for Linear First Order Systems Sec. 4.7 #3, 5, 9 March 5 22 [pdf] Variation of parameters for Linear Second Order ODEs Sec. 4.7 #11, 16, 23 March 7 23 [pdf] The Laplace Transform Sec. 5.1 # 7, 10, 17, 25 March 9 24 [pdf] 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 25 [pdf] The Inverse Laplace Transform Sec. 5.3 # 9, 17, 23 March 21 26 [pdf] Solving Initial Value Problems with Laplace Transforms Sec. 5.4 # 3, 11, 15 March 23 27 [pdf] Discontinuous Functions and Periodic Functions Sec. 5.5 # 5, 9, 13, 23 March 26 N/A Midterm 2 N/A March 28 28 [pdf] Impulse Functions Sec. 5.7 # 1, 3, 5 March 30 29 [pdf] Convolution Integrals and Their Applications Sec. 5.8 # 5, 9, 19 April 2 30 [pdf] Systems of First Order Linear ODEs: Definitions and Examples Sec. 6.1 # 2, 5, 9 April 4 31 [pdf] Basic Theory of First Order Linear Systems Sec. 6.2 #  5, 9, 11 April 6 32 [pdf] Homogeneous Linear Systems with Constant Coefficients Sec. 6.3 #  3, 7, 9 April 9 33 [pdf] Homogeneous Linear Systems with Constant Coefficients: Complex Eigenvalues Sec. 6.4 # 1, 5, 7 April 11 34 [pdf] Fundamental Matrices and the Exponential of a Matrix - I Sec. 6.5 # 3, 5, 7 April 13 35 [pdf] Fundamental Matrices and the Exponential of a Matrix - II Sec. 6.5 # 9, 11, 17 April 16 36 [pdf] Nonhomogeneous Linear Systems Sec. 6.6 # 1, finish example on Slide 5. April 18 37 [pdf] Autonomous Nonlinear Systems and Stability Sec. 7.1 # 3, 5, 7, 9, 11 April 20 38 [pdf] Almost Linear Systems Sec. 7.2 # 5(a,b,c), 7(a,b,c), 13(a,b,c), 19(a,b,c) April 23 39 [pdf] Competing Species Sec. 7.3 # 1, 3, 5 April 25 40 [pdf] Periodic Solutions and Limiting Cycles Sec. 7.5 # 3, 5, 11 April 27 41 [pdf] 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

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