Mathematics 53: Ordinary Differential Equations
Lectures: Each student needs to select one lecture section and one discussion section. There are two lecture sections.
Section 01: Monday/Wednesday/Friday, 10:00 AM -- 10:50 AM in Building 380, Room 380Y.
Section 02: Monday/Wednesday/Friday, 1:15 PM -- 2:05 PM in Building 380, Room 380Y.
Instructor: Simon Brendle, Room 382Z, brendle "at" math.stanford.edu
Office Hours: Monday/Wednesday/Friday, 2:05 PM -- 3:05 PM.
Discussion sections: There are six discussion sections, which are taught by Saran Ahuja, Chris Henderson, and Evan Warner. All discussion sections will begin on Thursday, September 26. There will be no discussion sections on Tuesday, September 24.
Saran Ahuja, Room 380M, sarana "at" math.stanford.edu.
Section 04: Tuesday/Thursday, 10:00 AM -- 10:50 AM, Room: 200-030
Section 05: Tuesday/Thursday, 11:00 AM -- 11:50 AM, Room: Meyer Forum
Office Hours: Monday/Wednesday/Friday 9:00 AM -- 10:00 AM.
Chris Henderson, Room 380N, chris "at" math.stanford.edu.
Section 06: Tuesday/Thursday, 1:15 PM -- 2:05 PM, Room: 380-380F
Section 08: Tuesday/Thursday, 2:15 PM -- 3:05 PM, Room: Herrin 195 (please note the room change)
Office Hours: Monday/Wednesday 11:00 AM -- 12:30 PM.
Evan Warner, Room 380M, ebwarner "at" math.stanford.edu.
Section 03: Tuesday/Thursday, 9:00 AM -- 9:50 AM, Room: 380-381T
Section 07: Tuesday/Thursday, 10:00 AM -- 10:50 AM, Room: 380-380W
Office Hours: Tuesday/Thursday 11:00 AM -- 12:30 PM.
You are welcome to visit any TA during office hours (not just the one teaching your section).
Course description: This is a first course in Ordinary Differential Equations. The material includes separation of variables; integrating factors and exact differential equations; systems of linear differential equations; eigenvalues and eigenvectors; the method of variation of parameters; the Laplace transform; systems of nonlinear differential equations, and conservation laws. This is a "cookbook"-style course, and the main focus will be on doing computations. If you are interested in a proof-oriented course on ordinary differential equations, you may want to consider taking Mathematics 53H.
Prerequisites: Mathematics 51 (or equivalent)
You should be familiar with eigenvalues and eigenvectors. You should also know the standard tricks for computing integrals of functions of a single variable (integration by parts, substitution, etc), and be able to use them in practice.
Textbook: The text for the course is "Differential Equations: An Introduction to Modern Methods and Applications" (2nd edition), by Brannan and Boyce.
Class 1 (Monday, September 23): Differential equations: what they are and why they are important; examples (heating and cooling of a building, falling objects, population models); direction fields; equilibrium solutions; qualitative behavior (Section 1.1)
Class 2 (Wednesday, September 25): Linear differential equations of first order with constant coefficients (Section 1.1)
Class 3 (Friday, September 27): Linear differential equations of first order with variable coefficients; integrating factors; examples (Section 1.2)
Class 4 (Monday, September 30): The method of separation of variables; examples of separable differential equations (Section 2.1)
Class 5 (Wednesday, October 2): More examples of separable equations (Section 2.1); the logistic growth model; autonomous differential equations (Section 2.4)
Class 6 (Friday, October 4): Homogeneous differential equations (cf. Problem 30 on page 51); examples; overview of the solutions methods discussed so far
Class 7 (Monday, October 7): Exact differential equations (Section 2.5); examples
Class 8 (Wednesday, October 9): Exact differential equations (continued); separable equations as special cases of exact differential equations
Class 9 (Friday, October 11): Existence and uniqueness theory for ordinary differential equations (Section 2.3); the interval of existence of a solution; analysis of equilibria
Class 10 (Monday, October 14): Systems of linear differential equations; coupled vs. uncoupled systems; applications to mixing problems; rewriting a differential equation of second order as a coupled system of two differential equations of first order; matrix notation (Section 3.2)
Class 11 (Wednesday, October 16): Existence and uniqueness theory for systems of linear ODE (Section 3.2); solving homogeneous systems with constant coefficients (Section 3.3); eigenvalues and eigenvectors
Class 12 (Friday, October 18): Solving homogeneous systems with constant coefficients (continued); examples; phase space plots; asymptotic behavior of solutions
Class 13 (Monday, October 21): Fundamental systems and Wronskian determinant (Section 3.3); 2x2 matrices with complex eigenvalues (Section 3.4); how to solve systems of ODE when the coefficient matrix has complex eigenvalues
Class 14 (Wednesday, October 23): How to solve systems of ODE when the coefficient matrix has complex eigenvalues (continued); expressing the solution in real form
Class 15 (Friday, October 25): More examples involving matrices with complex eigenvalues; matrices with repeated eigenvalues (Section 3.5)
Class 16 (Monday, October 28): More examples involving matrices with repeated eigenvalues; asymptotic behavior of solutions; sinks, sources, and saddles; classification in terms of trace and determinant of the coefficient matrix (diagram on page 190)
Class 17 (Wednesday, October 30): An example of a 3x3 system of linear differential equations; inhomogeneous 2x2 systems; variation of parameters formula (Section 4.7)
Class 18 (Friday, November 1): Variation of parameters formula (continued); any two solutions to a given inhomogeneous equation differ by a solution to the corresponding homogeneous equation
Class 19 (Monday, November 4): Second order linear differential equations; damped linear oscillations in mechanics (Section 4.4)
Class 20 (Wednesday, November 6): Linear oscillations in the presence of an external force; method of undetermined coefficients (Section 4.6); frequency response; gain function and phase shift; resonance (Section 4.6)
Class 21 (Friday, November 8): Laplace transform: definition and basic properties; a formula for the Laplace transform of the n-th derivative of a function (Sections 5.1 and 5.2)
Class 22 (Monday, November 11): A formula for the Laplace transform of t * f(t); examples; solving differential equations using the Laplace transform (Section 5.4); the inverse Laplace transform (Section 5.3)
Class 23 (Wednesday, November 13): Solving differential equations using the Laplace transform (continued); computing the inverse Laplace transform of a rational function using partial fractions
Class 24 (Friday, November 15): Computing the inverse Laplace transform of a rational function using partial fractions (continued); examples
Class 25 (Monday, November 18): Laplace transform and the transfer function; nonlinear systems of differential equations; existence and uniqueness theory (Section 3.6); autonomous systems and equilibria (Section 3.6, Example 1; Section 7.1)
Class 26 (Wednesday, November 20): Hamiltonian systems; the mathematical pendulum; conservation of energy (Section 7.1, Example 1)
Class 27 (Friday, November 22): The mathematical pendulum (continued); predator-prey models (Section 7.4)
Class 28 (Monday, December 2): Mathematical models for the spread of infections diseases: SIR models and the Kermack-McKendrick epidemic threshold theorem
Class 29 (Wednesday, December 4): Stability of equilibrium points; the linearization of a nonlinear system around an equilibrium point
Class 30 (Friday, December 6): The differential equations governing planetary motion; Kepler's laws
Exams and homework: The exams will focus on performing computations. The dates are as follows:
First midterm exam: Thursday, October 10, 7:00 PM -- 9:00 PM, Cubberley Auditorium.
Second midterm exam: Monday, November 4, 7:00 PM -- 9:00 PM, Annenberg Auditorium.
Final exam: Monday, December 9, 7:00 PM -- 9:30 PM, Braun Auditorium.
Homework will usually be assigned once a week, and should be turned in to your teaching assistant. Late submissions will only be accepted in truly exceptional circumstances. While you may discuss the homework problems with your peers, what you hand in must be your own work and not a joint project of several people. Your solutions should be readable and complete: this means that the grader should see a sufficient amount of explanations and details to give you full credit, even if the question only asks for a numerical answer.
Homework 1 (due on Tuesday, October 1, at 5 PM) Solutions
Homework 2 (due on Tuesday, October 8, at 5 PM) Solutions
Homework 3 (due on Tuesday, October 15, at 5 PM) Solutions
Homework 4 (due on Tuesday, October 22, at 5 PM) Solutions
Homework 5 (due on Tuesday, October 29, at 5 PM) Solutions
Homework 6 (due on Tuesday, November 5, at 5 PM) Solutions
Homework 7 (due on Tuesday, November 12, at 5 PM) Solutions
Homework 8 (due on Tuesday, November 19, at 5 PM) Solutions
Homework 9 (due on Tuesday, December 3, at 5 PM) Solutions
Grading: At the end of the quarter, we will compute a weighted average of your problem set scores (20%), your midterm scores (25% each), and your score on the final exam (30%). All homework scores will be taken into account when computing this weighted average. The final letter grades will be a assigned based on this weighted average.
Some general advice: Be sure to attend lectures. Skipping lectures on a regular basis all but guarantees less-than-stellar performance on the exams.
Take advantage of office hours, and ask lots of questions during class! We always welcome questions (both during class and during office hours), but it is your responsibility to let us know if an issue requires clarification.