MAT 22B (Summer Session I, 2017). Differential Equations

Instructor: Matthew Cha
Office hours: MSB 3139, MW 10:00am-11:00am and 4:00pm-5:00pm or by appointment
Lecture: Haring 2016, MWF 8:00am-9:40am
Email: mmcha "at" math "dot" ucdavis "dot" edu

Course description

This course is an introduction to ordinary differential equations (ODEs). Differential equations most often arise as mathematical models of real situations, which is why scientists and engineers, as well as mathematicians, study them. In this course we will learn how to solve first and second order linear ODEs, by multiple methods, and also learn how to solve systems of first order linear ODEs.

The prerequisite for this course is a thorough knowledge of calculus and course 22A or 67 with C- or above. The textbook is W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 10th ed., (Hoboken, NJ: John Wiley & Sons, Inc. 2012). We will cover chapters 1, 2, 3, 6 and 7. We will follow the syllabus posted on the math department website somewhat closely.

There will be six weekly homework sets, due in class (or in the MSB first floor drop box for this course before midnight) on Fridays. Please note that the front doors of MSB are locked at 7pm. Homework sets will typically be assigned the Friday before they are due. Solutions will be posted the day after homework is collected. Late homework will not be accepted. Students are allowed to discuss the homework assignments among themselves, but are expected to turn in their own work — copying someone else's is not acceptable. Homework scores will contribute 30% to the final grade. The lowest homework score will be dropped.

There will be one midterm, set for 8:00a-8:50a Friday, July 14. The final is scheduled for 8:00a Friday, Aug 4; this is the last scheduled day of the course at the regular lecture time. There will be no makeup tests.

The grade for the course will be calculated based on the following percentages: homework 30%, midterm 30%, final 40%. Grades will be posted on Canvas.

Syllabus (subject to modification)

26 Jun 17 §1.1. Mathematical modeling; direction fields
         Simple pendulum, see problem §1.3.29
§1.2. Solving first order linear constant coefficient ODEs
§1.4. History [1]
HW1 (due Mon Jul 3). §1.1: 3,7,15,24; §1.2: 1a,2a,13; §1.3: 1,4,12,30
         [solutions]
28 Jun 17 §1.3. Linear ODEs
§2.1. First order linear equations; integrating factors
§2.2. Seperable equations
30 Jun 17 §2.3. Applications of first order ODEs
         conduction, mixing, escape velocity, black holes [2]
§2.4. Existence and uniqueness; linear vs nonlinear equations
HW2 (due Fri Jul 7). §2.1: 9,13,18,30; §2.2: 3,12,23,30; §2.3: 8,12,16,31ab; §2.4: 2,8,14,21,25,32
Extra credit 1 [solutions]
03 Jul 17 §2.5. Autonomous equations and population dynamics
§2.7. Euler's method for numerical approximation
05 Jul 17 §2.8. Picard's method for existence and uniqueness of solutions
         sequences of functions, convergence, mathematial induction
07 Jul 17 Second order differential operators [3]
§3.1. Homogeneous equations with constant coefficients and the characteristic equation
§3.3. Complex roots and Euler's formula
§3.4. Repeated roots
HW3 (due Fri Jul 14). §2.5: 5,10,20; §2.7: 11a(t=0.5),20; §2.8: 13,19;
§3.1: 14,17,23; §3.2: 4,34,39; §3.3: 9,20,29; §3.4: 13,12; §3.7: 5,7

Extra credit 2 [solutions]
Practice midterm
10 Jul 17 §3.7. Harmonic oscillator; simple and damped
§3.2. Linear independence of solutions and Wronskian
12 Jul 17 §3.2 Wronskian and Abel's theorem
         Midterm will cover sections 1.1-4, 2.1-5, 3.1, 3.3-4, 3.7
         Practice MT [problem 2]
14 Jul 17 Midterm [solutions]
Nonhomogeneous Equations
§3.5 Method of undetermined coefficients
HW4 (due Fri Jul 21). §3.4: 20,31; §3.5: 8, 16, 31, 35
[solutions]
17 Jul 17 §3.5 Method of undetermined coefficients (cont.)
§3.6 Variation of parameters
19 Jul 17 §3.8 Forced harmonic oscillators
         resonance, amplitude modulation
21 Jul 17 HW5 (due Fri Jul 28). §3.6: 11,17,25; §3.8: 12,15;
         §7.1: 6, 10; §7.2: 8,21cd,23; §7.3: 14,18,27,33;

[solutions]
24 Jul 17 §7.1 Systems of first order linear equations
§7.2 Review of linear algebra and matrices
§7.3 Eigenvalue and eigenvector problem
26 Jul 17 §7.2 Inner product spaces
§7.7 Matrix exponential: diagonalizable matrices
§7.5 Homogeneous linear systems with constant coefficients
28 Jul 17 Matrix exponential: two by two matrices
§7.5 Phase portrait and equilibrium
§7.6 Complex eigenvalues
§7.8 Non-diagonalizable matrix
HW6 (due Wed Aug 2). §7.5: 3a,24,31; §7.6: 9,13,26; §7.7: 3,11,17; §7.8: 15,16
[solutions]
Practice final
25 Jul 16 §6.1 Laplace transform
§6.2 Solution of initial value problems
§6.3 Step functions
§6.4 Discontinuous forcing functions
suggested problems. §6.1: 1,5,22; §6.2: 7,11,24;
[solutions]
4 Aug 17 Final [solutions]

Suggested reading

[1] R. P. Feynman, "Surely You're Joking, Mr. Feynman!": Adventures of a Curious Character, as told to R. Leighton, edited by E. Hutchings (New York: Bantam Books 1986).
[2] P.-S. de Laplace, "Proof that the attractive power of a heavenly object can be so large that light cannot be emitted from it", translated from Allgemeine geographische Ephemeriden, Bd I (1799).
[3] D. A. Meyer "Solving second order linear ODEs with constant coefficients using differential operators and their inverses", lecture notes, 2007.

Last modified: 04 Jul 17.