# MATH 211: Ordinary Differential Equations and Linear Algebra - Summer 2009

Instructor: Christian Bruun
Office: Herman Brown 42
Tel: (713) 348-8304
Email: cbruun@rice.edu
Time: MWF 10 am – 12 pm
Location: SH 560

Syllabus (PDF)

Announcements

• June 1: We will have office hours TTh 10 am – 12 pm in Herman Brown 42.

• June 17: Our classroom has been switched to Sewall Hall 560.

• June 19: The first midterm exam will be distributed in class. It is a 2 hour take home exam. Please return it in class on Monday, June 22. There is a sample exam available online.

• July 3: No class today due to University holiday.

• July 6: No class today.

• June 10: The first midterm exam will be distributed in class. It is a 2 hour take home exam. Please return it in class on Monday, July 13. There is a sample exam available online.

Schedule

• June 1:

• Sections: 1.1, 1.2, 1.3

• Important Concepts: models, derivative, antiderivative, integration, initial value problem, initial condition

• Homework: Due Wednesday, June 3

• 1.1: 1, 2, 3, 4, 7, 11

• 1.2: 10, 11, 13

• 1.3: 17, 20, 26

• June 3:

• Sections: 2.1, 2.2, 2.3

• Important Concepts: ordinary differential equations, partial differential equations, independent variable, normal form, solutions of ODEs, initial value problem, interval of existence, direction field, equilibrium solution, separable equation, implicit solutions, linear motion

• Homework: Due Friday, June 5

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

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

• 2.3: 4, 5, 9

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.

• June 5:

• Sections: 2.4, 2.5, 2.6

• Important Concepts: homogeneous/inhomogeneous equations, integrating factor, variation of parameters, mixing problems, differential forms

• Homework: Due Monday, June 8

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

• 2.5: 1, 3, 12

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

• June 8:

• Sections: 2.7, 2.8, 2.9

• Important Concepts: existence/uniqueness of solutions, interval of existence, deterministic system, existence/uniqueness theorems, geometric interpretation of uniqueness, continuity with respect to initial conditions, sensitivity of solutions to initial conditions, autonomous equations, equilibrium points/solutions, nonequilibrium solutions, phase line, stability, asymptotically stable, unstable, first derivative test

• Homework: Due Wednesday, June 10

• 2.7: 1, 2, 3, 11, 12, 23, 27, 29

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

• June 10:

• Sections: 3.1, 3.2, 3.3, 3.4

• Important Concepts: population models, birth/death/reproduction rate, Malthusian model, logistic model, logistic equation, natural reproduction rate, current, component laws

• Homework: Due Friday, June 12

• 3.1: 1, 3, 10, 13, 17

• 3.3: 1, 2

• 3.4: 5, 13, 15, 21

• June 12:

• Sections: 4.1, 4.2

• Important Concepts: second-order differential equations, linear equations, forcing term, homogeneous equations, spring equilibrium, damping constant, spring constant, natural frequency, period, linear combination, linear dependence, fundamental set of solutions, Wronskian, planar systems, phase plane

• Homework: Due Monday, June 15

• 4.1: 1, 2, 3, 9, 10, 13, 14, 18, 22

• 4.2: 1, 2, 4

• June 15:

• Sections: 4.3, 4.4

• Important Concepts: characteristic equation/polynomial, characteristic root, harmonic motion, forcing term, amplitude, phase, damped harmonic motion, underdamped, overdamped, critically damped

• Homework: Due Wednesday, June 17

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

• 4.4: 11, 13

• June 17:

• Sections: 4.5, 4.6, 4.7

• Important Concepts: inhomogeneous equations, method of undetermined coefficients, variation of parameters, forced harmonic motion, beats, transfer function, gain, transient, lumping parameters

• Homework: Due Friday, June 19

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

• 4.6: 1, 3, 6, 10, 13

• 4.7: 8, 9, 27

• June 19:

• First midterm exam distributed in class: This is a 2 hour take home exam. Please return it in class on Monday, June 22.

• Sections: 5.1, 5.2, 5.3

• Important Concepts: Laplace transform, piecewise continuous functions, inverse transforms, potential function

• Homework: Due Monday, June 22

• 5.1: 3, 7, 10, 28, 29

• 5.2: 3, 5, 7, 19, 21

• June 22:

• Sections: 5.4, 5.5, 5.6, 5.7

• Important Concepts: discontinuous forcing terms, interval function, Heaviside function, square wave, delta function, impulse response function, convolutions

• Homework: Due Wednesday, June 24

• 5.2: 23, 25, 27, 39

• 5.3: 1, 3, 7, 13, 23, 25, 33

• 5.4: 3, 9, 11, 19, 35

• 5.5: 1, 11, 15, 17, 21

• June 24:

• Sections: 6.1, 6.2, 6.3, 6.4

• Important Concepts: numerical solution methods, Euler's method, step size, error, Runge-Kutta methods, variable-step solvers, stiff solvers

• Homework: Due Friday, June 26

• 6.1: 2, 3, 10, 17, 18

• 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 (scroll down a few pages to the table giving a description of each solver and their uses). 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 an example function file for problem 6.1.10. This defines the function f(t,y)=48y-y^2. You can run this with the 'eul.m' function (over the interval [0,2] with y(t_0)=1 and step-size 0.04) using the command line:

[tout, yout]=eul(@(t,y) myode1(t,y),[0,2],1,0.04);

To approximate a different function, edit the function in 'myode1.m'.

• June 26:

• 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: Due Monday, June 29

• 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

• June 29:

• Sections: 7.4, 7.5

• Important Concepts: homogeneous/inhomogeneous systems, nontrivial solution, nullspace, span, subspace, linear dependence/independence, basis, dimension

• Homework: Due Wednesday, July 1

• 7.4: 3, 4, 7, 15, 23, 27

• 7.5: 3, 5, 13, 17, 19, 27, 29

• July 1:

• Sections: 7.6, 7.7

• Important Concepts: nonsingular/singular matrices, matrix inverse, determinant

• Homework: Due Wednesday, July 8

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

• 7.7: 9, 19, 25, 27, 30, 31, 45

• July 3: University Holiday

• No class today

• July 6:

• No class today

• July 8:

• Sections: 8.1, 8.2, 8.3

• Important Concepts: SIR model of an epidemic, autonomous systems, reduction of higher-order equations to first-order system, predator-prey systems, parametric plot, phase plane, vector field, existence and uniqueness of solutions of systems, equilibrium point, nullcline

• Homework: Due Friday, July 10

• 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

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.

• July 10:

• Second midterm exam distributed in class: Please return it in class on Monday, July 13.

• Sections: 8.4, 8.5

• Important Concepts: linear systems, homogeneous/inhomogeneous systems, linear independence/dependence of solutions of a linear system, Wronskian of a set of solutions

• Homework: Due Monday, July 13

• 8.4: 7, 9, 19, 21, 25

• 8.5: 1, 7, 11, 17, 19, 23

• July 13:

• Sections: 9.1, 9.2, 9.3

• 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

• Homework: Due Wednesday, July 15

• 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

• July 15:

• Sections: 9.4, 9.5, 9.6

• Important Concepts: trace-determinant plane, geometric/algebraic multiplicity of an eigenvector, exponential of a matrix, truncation, generalized eigenvectors/solutions

• Homework: Due Friday, July 17

• 9.4: 1, 3, 5, 17, 19

• 9.5: 3, 5, 9, 11, 19, 21, 23, 35

• 9.6: 1, 3, 7, 13, 17, 27

• July 17:

• Sections: 9.7, 9.8, 9.9

• Important Concepts: qualitative analysis of linear equations, higher-order linear equations, inhomogeneous linear systems, fundamental matrix

• Homework: Due Monday, July 20

• 9.7: 1, 3, 9, 11

• 9.8: 1, 3, 15, 23, 25, 29

• 9.9: 1, 3, 15, 27

• July 20:

• Final exam distributed

• Sections: TBD

• Important Concepts: TBD

• July 27:

• Final exam due by 5 pm