# MATH 211: Ordinary Differential Equations and Linear Algebra - Summer 2008

Instructor: Christian Bruun
Office: Herman Brown 42
Tel: (713) 348-8304
Email: cbruun@rice.edu
Time: MTWRF 1 pm – 4:05 pm
Location: GRB W211

Syllabus (PDF)

Announcements

• May 13: We will have office hours MTWRF 11 am – 12 pm in Herman Brown 42.

• May 19: First in-class quiz.

• May 21: Second in-class quiz.

• May 23: Third in-class quiz.

• May 28: Fourth in-class quiz.

• May 30: The final exam will be distributed.

Schedule

• May 13:

• Sections: 1.1, 1.2, 1.3, 2.1, 2.2

• Important Concepts: models, derivative, integral, ordinary differential equation, independent variable, normal form, solution, initial value problem, interval of existence, direction field, equilibrium solution, separable equation

• Homework: Due Thursday, May 15

• 1.1: 1, 2, 3, 4, 7, 11

• 1.2: 10, 11, 13

• 1.3: 17, 20, 26

• 2.1: 2, 4, 7, 9, 14, 17, 18, 24, 31, 32, 39

• 2.2: 1, 3, 8, 13, 16, 17, 19, 23, 36

Note: You will need to use some kind of mathematical plotter for some problems. You may use anything you have available, but I suggest you use MATLAB, along with the dfield applet. You can find this here. Please see me if you have trouble getting this to work, or otherwise need some help.

• May 14:

• Sections: 2.3, 2.4, 2.5

• Important Concepts: linear motion, air resistance, homogeneous/inhomogeneous equations, integrating factor, variation of parameters

• Homework: Due Friday, May 16

• 2.3: 4, 5, 9

• 2.4: 2, 7, 10, 14, 15, 19, 30, 31, 37, 38

• 2.5: 1, 3, 12

• May 15:

• Sections: 2.6, 2.7

• Important Concepts: differential forms, existence/uniqueness of solutions, deterministic system

• Homework: Due Monday, May 19

• 2.6: 1, 3, 11, 13, 17, 19, 23, 25, 26, 31, 33, 35, 44

• 2.7: 1, 2, 3, 11, 12, 23, 27, 29

• May 16:

• Sections: 2.9, 3.1, 3.3, 4.1

• Important Concepts: autonomous equations, equilibrium solutions, phase line, stability, Malthusian model, logistic model, second-order differential equations, forcing term, homogeneous equations, spring model, natural frequency, period, linear dependence, fundamental set of solutions, Wronskian

• Homework: Due Tuesday, May 20

• 2.9: 2, 3, 5, 8, 10, 11, 15, 17, 23, 27

• 3.1: 1, 3, 10, 13, 17

• 3.3: 1, 2

• 4.1: 1, 2, 3, 9, 10, 13, 14, 18, 22

• May 19:

• In-class quiz

• Sections: 4.2, 4.3, 4.4, 4.5

• Important Concepts: planar system, phase plot, characteristic equation, distinct real roots/complex roots/repeated roots, harmonic motion, damping, forcing term, amplitude, phase, underdamped, overdamped, critically damped, method of undetermined coefficients

• Homework: Due Wednesday, May 21

• 4.2: 1, 2, 4

• 4.3: 3, 4, 5, 11, 13, 14, 18, 19, 27, 29, 30, 33, 38

• 4.4: 11, 13

• 4.5: 1, 3, 5, 7, 19, 23, 25, 27, 31, 32

• May 20:

• Sections: 4.6, 4.7, 6.1

• Important Concepts: variation of parameters, forced harmonic motion, driving frequency, beats, mean frequency, transfer function, gain, Euler's method, error

• Homework: Due Thursday, May 22

• 4.6: 1, 3, 5, 6, 10, 13

• 4.7: 8, 9, 27

• 6.1: 2, 3, 10, 17, 18

• May 21:

• In-class quiz

• Sections: 6.2, 6.3, 6.4

• Important Concepts: second-order Runge-Kutta, fourth-order Runge Kutta, variable-step solvers, stiff solvers

• Homework: Due Tuesday, May 27

• 6.2: 1, 3, 7, 20, 21

• 6.3: 11

• 6.4: 1, 2, 3

Note: You will need computer implementations of some numerical solvers for a few problems. Again, you may use whatever program you have available that has these. There are MATLAB implementations of second- and fourth-order Runge-Kutta methods available here. Also, you may want to use the ODE solvers built in to MATLAB. You can find information about those here, along with some examples. Some ODE solvers are simpler to use than others; if you use MATLAB it may be a little daunting. Please email or see me if you need help with something.

Here is a Maple worksheet with some plots for problem 4.7.8.

• May 22:

• Sections: 7.1, 7.2, 7.3

• Important Concepts: vectors, matrices, matrix sum, matrix multiplication, scalar multiplication, transpose, solution set, parametric representation, free parameter, elimination, back-solving, pivot, row echelon form, row operations, consistency, reduced row echelon form

• Homework: Homework: Due Tuesday, May 27

• 7.1: 9, 33, 34, 35, 36, 37, 49, 50, 51

• 7.2: 3, 13, 21, 22, 36

• 7.3: 1, 2, 3, 4, 19, 20, 35, 36

• May 23:

• In-class quiz

• Sections: 7.4, 7.5, 7.6, 7.7

• Important Concepts: homogeneous systems, inhomogeneous systems, nontrivial solutions, nullspace, span, subspace, linear independence/dependence, basis, dimension, square matrix, nonsingular/singular, invertible, determinant

• Homework: Due Wednesday, May 28

• 7.4: 3, 4, 7, 15, 23

• 7.5: 3, 5, 13, 17, 19, 27, 29

• 7.6: 3, 4, 5, 13, 15, 20, 21, 23, 29, 34

• 7.7: 9, 19, 25, 27, 30, 31, 45

• May 26: Memorial Day

• No class today

• May 27:

• Sections: 8.1, 8.2, 8.3, 8.4, 8.5

• Important Concepts: autonomous systems, reduction of higher-order equations to first-order system, predator-prey system, parametric plot, phase plane, vector field, existence and uniqueness of solutions of systems, equilibrium point, nullcline, linear systems, homogeneous/inhomogeneous systems, linear independence/dependence of solutions of a linear system, Wronskian of a set of solutions

• Homework: Due Thursday, May 29

• 8.1: 3, 5, 9, 15, 16, 23, 24

• 8.2: 1, 5, 9, 13, 14, 21, 26

• 8.3: 3, 5, 7, 9, 15

• 8.4: 7, 9, 19, 21, 25

• 8.5: 1, 7, 11, 17, 19, 23

Note: Some problems will require a plotter for constructing vector field plots or phase plane solutions. There is an applet similar to dfield, called pplane, available here.

• May 28:

• In-class quiz

• Sections: 9.1, 9.2, 9.3, 9.4, 9.5

• Important Concepts: linear systems, eigenvalue, eigenvector, eigenspace, characteristic polynomial, planar systems, phase plane portrait, saddle points, separatrix, node, sink, source, center, spiral sink, spiral source, trace-determinant plane, geometric/algebraic multiplicity of an eigenvector

• Homework: Due Friday, May 30

• 9.1: 1, 3, 9, 14, 17, 19, 23, 25

• 9.2: 1, 3, 7, 9, 16, 29, 31, 43

• 9.3: 11, 13, 17, 19, 21

• 9.4: 1, 3, 5, 17, 19

• 9.5: 3, 5, 9, 11, 19

• May 29:

• Sections: 9.5, 9.6, 9.7

• Important Concepts: exponential of a matrix, truncation, generalized eigenvectors, stability, variation of parameters, linearization of nonlinear system, Jacobian

• Homework: This homework will not be collected, though I strongly suggest you work on them.

• 9.5: 21, 23, 35

• 9.6: 1, 3, 7, 13, 17, 27

• 9.7: 1, 3, 9, 11

• 9.8: 1, 3, 15, 23, 25, 29

• 9.9: 1, 3, 15, 27

• May 30:

• Final exam distributed

• Sections: 9.8, 9.9

• June 3:

• Final exam due by 5 pm