Instructor: Matthew Cha
Office hours: MSB 3139, W 10:00am-11:00am and 4:00pm-5:00pm, Th 2:00pm-4:00pm, or by appointment
Lecture: Olson 158, MWF 2:10pm-3:50pm
Email: mmcha "at" math "dot" ucdavis "dot" edu
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, 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. 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 2:10pm-3:00pm Monday July 11. The final is scheduled for 2:10pm Friday July 29; 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 smartsite.
| 20 Jun 16 |
§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 27 Jun 16). §1.1: 3,7,15,24; §1.2: 1a,2a,13; §1.3: 1,4,12,30 [solutions] |
| 22 Jun 16 |
§1.3. Linear ODEs §2.1. First order linear equations; integrating factors §2.2. Seperable equations |
| 24 Jun 16 |
§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 1 Jul 16). §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] |
| 27 Jun 16 |
§2.5. Autonomous equations and population dynamics §2.7. Euler's method for numerical approximation |
| 29 Jun 16 |
§2.8. Picard's method for existence and uniqueness of solutions sequences of functions, convergence, mathematial induction |
| 1 Jul 16 |
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 Mon 11 Jul 16). §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] |
| 4 Jul 16 | Independence Day! (No class) |
| 6 Jul 16 |
§3.7. Harmonic oscillator; simple and damped §3.2. Fundamental set of solutions to linear homogeneous equations linear independence, Wronskian, Abel's theorem Practice midterm |
| 8 Jul 16 |
Nonhomogeneous equations §3.5 Method of undetermined coefficients Midterm review |
| 11 Jul 16 |
Midterm [solutions] §3.4-5 Reduction of order HW4 (due Fri 15 Jul 16). §3.4: 20,31; §3.5: 8, 16, 31, 35 Extra credit 3 [solutions] |
| 13 Jul 16 |
§3.6 Variation of parameters §3.8 Forced harmonic oscillators and resonance |
| 15 Jul 16 |
§7.1 Systems of first order linear equations §7.2 Review of linear algebra and matrices §7.3 Eigenvalue and eigenvector problem HW5 (due Fri 22 Jul 16). §3.6: 3,11,17,25,26; §3.8: 12,15; §7.1: 6,7,10,14; §7.2: 2,8,21cd,23; §7.3: 8,14,18,27,31,33 Extra credit 4 [solutions] |
| 18 Jul 16 |
§7.2 Inner product spaces §7.7 Diagonalization of self-adjoint matrices |
| 20 Jul 16 |
§7.7 Matrix exponential: diagonalizable matrices §7.5 Homogeneous linear systems with constant coefficients |
| 22 Jul 16 |
Matrix exponential: two by two matrices §7.5 Phase portrait and equilibrium §7.6 Complex eigenvalues §7.8 Repeated eigenvalues HW6 (due Wed 27 Jul 16). §7.5: 3a,24,31; §7.6: 9,13,26; §7.7: 3,11,17; §7.8: 8,12a,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 §6.6 The convolution integral suggested problems. §6.1: 1,5,22; §6.2: 7,11,24,29; §6.6: 3,13 |
| 27 Jul 16 |
Partial differential equations Quantum harmonic oscillator Final review |
| 29 Jul 16 | Final [solutions] |
| [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. |