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 260, Room 113.
Section 02: Monday/Wednesday/Friday, 2:15 PM -- 3:05 PM in Building 380, Room 380C.
Instructor: Simon Brendle, Room 382Z, brendle "at" math.stanford.edu
Office Hours: Monday/Wednesday/Friday, 3:05 PM -- 4:00 PM.

Discussion sections: There are six discussion sections, which are taught by Ian Petrow, Maksym Radziwill, and David Ren. All discussion sections will begin on Thursday, January 10. There will be no discussion sections on Tuesday, January 8.

Ian Petrow, Room 381B, ipetrow "at" math.stanford.edu.
Section 07: Tuesday/Thursday, 10:00 AM -- 10:50 AM, Room: Organic Chemistry 103.
Section 08: Tuesday/Thursday, 2:15 PM -- 3:05 PM, Room: 320-106.
Office Hours: Tuesday/Thursday, 3:15 -- 4:45 PM.

Maksym Radziwill, Room 381G, maksymr "at" math.stanford.edu.
Section 03: Tuesday/Thursday, 11:00 AM -- 11:50 AM, Room: McCullough 126.
Section 05: Tuesday/Thursday, 2:15 PM -- 3:05 PM, Room: 380-380F.
Office Hours: Wednesday 11:00 AM -- 2:00 PM.

David Ren, Room 381F, weiluo "at" math.stanford.edu.
Section 04: Tuesday/Thursday, 10:00 AM -- 10:50 AM, Room: Meyer Library, Forum Room 124.
Section 06: Tuesday/Thursday, 1:15 PM -- 2:05 PM, Room: Herrin T185.
Office Hours: Monday 11:00 AM -- 2:00 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.

Tentative schedule:
Class 1 (Monday, January 7): 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, January 9): Linear differential equations of first order with constant coefficients (Section 1.1)
Class 3 (Friday, January 11): Linear differential equations of first order with variable coefficients; integrating factors; examples (Section 1.2)
Class 4 (Monday, January 14): The method of separation of variables; examples of separable differential equations (Section 2.1)
Class 5 (Wednesday, January 16): More examples of separable equations (Section 2.1); the logistic growth model; autonomous differential equations (Section 2.4)
Class 6 (Friday, January 18): Homogeneous differential equations (cf. Problem 30 on page 51); examples; overview of the solutions methods discussed so far
Class 7 (Wednesday, January 23): Exact differential equations (Section 2.5); examples
Class 8 (Friday, January 25): Exact differential equations (continued); separable equations as special cases of exact differential equations
Class 9 (Monday, January 28): Existence and uniqueness theory for ordinary differential equations (Section 2.3); the interval of existence of a solution; analysis of equilibria
Class 10 (Wednesday, January 30): 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 (Friday, February 1): 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 (Monday, February 4): Solving homogeneous systems with constant coefficients (continued); examples; phase space plots; asymptotic behavior of solutions
Class 13 (Wednesday, February 6): 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 (Friday, February 8): How to solve systems of ODE when the coefficient matrix has complex eigenvalues (continued); expressing the solution in real form
Class 15 (Monday, February 11): More examples involving matrices with complex eigenvalues; matrices with repeated eigenvalues (Section 3.5)
Class 16 (Wednesday, February 13): 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 (Friday, February 15): An example of a 3x3 system of linear differential equations; inhomogeneous 2x2 systems; variation of parameters formula (Section 4.7)
Class 18 (Wednesday, February 20): Variation of parameters formula (continued); any two solutions to a given inhomogeneous equation differ by a solution to the homogeneous equation
Class 19 (Friday, February 22): Second order linear differential equations; damped linear oscillations in mechanics (Section 4.4)
Class 20 (Monday, February 25): Linear oscillations in the presence of an external force; method of undetermined coefficients (Section 4.6)
Class 21 (Wednesday, February 27): Linear oscillations in the presence of an external force (continued); frequency response; gain function and phase shift; resonance (Section 4.6)
Class 22 (Friday, March 1): 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 23 (Monday, March 4): 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 24 (Wednesday, March 6): Solving differential equations using the Laplace transform (continued); computing the inverse Laplace transform of a rational function using partial fractions
Class 25 (Friday, March 8): 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 (Monday, March 11): Hamiltonian systems; the mathematical pendulum; conservation of energy (Section 7.1, Example 1)
Class 27 (Wednesday, March 13): The mathematical pendulum (continued); predator-prey models (Section 7.4)
Class 28 (Friday, March 15): The linearization of a nonlinear system (Section 7.2); the differential equations governing planetary motion; Kepler's first law

Exams and homework: The exams will focus on doing computations. The dates are as follows:
First midterm exam: Tuesday, January 29, 7:00 PM -- 9:00 PM, Cubberley Auditorium.
Second midterm exam: Thursday, February 21, 7:00 PM -- 9:00 PM, Cubberley Auditorium.
Final exam: Monday, March 18, 7:00 PM -- 9:00 PM (Room TBA).

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, January 15, at 5 PM) Solutions
Homework 2 (due on Tuesday, January 22, at 5 PM) Solutions
Homework 3 (due on Tuesday, January 29, at 5 PM) Solutions
Homework 4 (due on Tuesday, February 5, at 5 PM) Solutions
Homework 5 (due on Tuesday, February 12, at 5 PM)
Homework 6 (due on Tuesday, February 19, at 5 PM)

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.