Math 255, Section 104: Ordinary Differential Equations (Fall 2010)

Instructor: Dr. Jessica M. Conway.
Lectures: MWF 1-2pm, Henn 200.
Office hours: Mathematics Annex, Room 1110 - Wednesdays 4-6pm + by appointment.
OFFICE HOURS DURING FINAL EXAMS (DEC 6-20): Tuesdays/Thursdays from 3-5pm; Saturday Dec 18/Sunday Dec 19 from 4-7pm.
Email: conway (at) math (dot) ubc (dot) ca OR math255s104.fall2010 (at) gmail (dot) com
Phone: (604)822-6754
The Mathematics Department offers Drop in tutoring, ODEs included!
Schedule and locations available at http://www.math.ubc.ca/Ugrad/ugradTutorials.shtml.
Printable course outline available HERE.

Text: Boyce and Diprima, Elementary differential equations and boundary value problems, 9th edition.
We will cover chapters 1-3, 6, 7, and 9.
Note: If you have instead an 8th edition of the text, that's fine. Problems and readings for the 8th edition are also provided below.

ANNOUNCEMENTS:

  • (18/12/2010) Solutions to December 2009 Final posted (see 'Exams' section below).
  • (7/12/2010) Final review scheduled (see 'Exams' section below).
  • (23/11/2010) Final review scheduling survey posted (see 'Exams' section below).
  • (17/11/2010) Problems from today's problem session posted (see 'Documents' section below).
  • (13/11/2010) Table of Laplace transforms posted (see 'Exams' section below). This same table will be included with your midterm 3.
  • (13/11/2010) Review problems for midterm 3 posted (see 'Exams' section below).
  • (12/11/2010) Practice midterms for midterm 3 posted (see 'Exams' section below).
  • (8/11/2010) Handout with a couple of eigenvalue/eigenvector examples posted (see 'Documents' section below).
  • (8/11/2010) Handout summarizing linear systems in the plane posted (see 'Documents' section below).
  • (3/11/2010) Problems from today's problem session posted (see 'Documents' section below).
  • (3/11/2010) Practice problem on discontinuous forcing functions using convolution integrals posted (see 'Documents' section below).
  • (1/11/2010) Midterm 2 solutions posted posted (see 'Exams' section below).
  • (1/11/2010) Practice problem on discontinuous forcing functions posted (see 'Documents' section below).
  • (27/10/2010) Summary of Partial Fractions handout posted (see 'Documents' section below).
  • (18/10/2010) Review problems for midterm 2 posted (see 'Exams' section below).
  • (13/10/2010) Practice problem on forced oscillations posted (see 'Documents' section below).
  • (13/10/2010) Reduction of order example updated with more details (see 'Documents' section below).
  • (8/10/2010) Additional example using variation of parameters posted (see 'Documents' section below).
  • (6/10/2010) Table of trial expressions for the method of undetermined coefficients posted (see 'Documents' section below).
  • (29/09/2010) Additional example using reduction of order posted (see 'Documents' section below).
  • (29/09/2010) Review problems for midterm 1 posted (see 'Exams' section below).
  • (27/09/2010) Problems from first problem session posted (see 'Documents' section below).
  • (24/09/2010) Practice midterm posted with solutions (see 'Exam Dates' section below).
  • (24/09/2010) Monday's problem session will be in MATX 1102.

    Exam Dates:

  • Midterm test 1: Monday, October 4th
    On material up to and including lecture on Wed Sept 22 (up to and including section 3.1 of textbook).
    Practice midterm AND solutions (in problem 2, don't worry about non-zero right hand side).
    Review problems: 1.1: 15-20, 23, 24; 1.2: 1,3, 16; 2.1: 9, 15, 17-20, 30, 31; 2.2: 1-6, 31ab, 32ab, 36ab; 2.3: 2, 6, 10, 13, 16, 24; 2.4: 1, 3, 6, 22; 2.5: 7, 19; 3.1: 3, 6, 10, 13, 17.
    Note that these are meant to be just a guide; if you're having trouble with a concept, do more problems!
    Midterm and solutions

  • Midterm test 2: Monday, October 25th
    On material in sections 3.1-3.8 of textbook (second part of lecture on Wed. Sept. 22 to first part of lecture on Mon. Oct. 18).
    Review problems: 3.1: 5, 14, 15; 3.2: 2, 17, 21; 3.3: 10, 12; 3.4: 8, 11, 13, 27, 29; 3.5: 7, 14, 15, 18, 20; 3.6: 4, 6, 10, 13, 16; 3.7: 12, 13, 17, 24, 27, 30; 3.8:6, 10, 12, 16, 17, 18.
    Midterm and solutions

  • Midterm test 3: Friday, November 19th
    On Laplace transforms and solving linear systems (up to undetermined vectors only). Note that sketching of phase planes will not be tested on this midterm.
    Practice midterm A AND solutions
    Practice midterm B AND solutions
    Review problems: 6.1: 1, 6; 6.2: 3, 5, 21, 26; 6.3: 6, 15, 23; 6.4: 3, 9, 16; 6.5: 5, 7; 6.6: 14, 16; 7.1: 5, 22; 7.5: 9, 16, 30; 7.6: 6, 13, 25, 29; 7.8: 3, 7, 15, 16; 7.9: 2, 3, 4.
    Table of Laplace transforms. This will be the table included with your midterm exam. Note that it's identical to the one in your textbook.
    Midterm and solutions

  • Final exam: December 20th, 8:30AM in LSK 200
    • Note: No notes, books or calculators will be allowed for in-class midterms or the Final Exam.
    Final Exam Review: Tuesday December 14th at 7pm-9pm, MATX 1100 AND Wednesday December 15th, 7-9pm, MATX 1100.
    NEW Laplace transforms table for the final available here.
    NEW Solutions to the December 2009 final. Please let me know immediately if you find any mistakes, to prevent any confusion on the part of your classmates.

    Grading

  • 45% from the midterms + homework assignments; 55% from the Final Exam.
    You must have a passing grade on the best nine of 11 homework assignments and pass the Final Exam to pass this course.
  • If you miss a midterm with a valid reason, then your term mark (45%) will be based on the other two midterms + homework assignments.
  • Term marks may be scaled up or down on a classwide basis, depending on performance on the final exam. This is to ensure fairness across all sections of the course.

    • Homework

  • due each week at the beginning of the Friday class. Problems to be handed-in may be selected from the indicated Suggested Problems, listed below in the lecture outline.
    • Homework 1, due Sept 17th 2010:
    9th edition: p.15: 1(a), 4, 8, 13, 18; p.24: 18, 20; p.39: 5, 24.
    OR 8th edition: p.15: 1(a), 4, 8, 13, 18; p.24: 18, 20; p.39: 5, 24.
    SOLUTIONS here

    Homework 2, due Sept 24th 2010:
    9th edition: p.47: 34; p.59: 32; p.75: 3; p.88: 15,22; p.99: 13.
    OR 8th edition: p.47: 34; p.59: 32; p.75: 3; p.88: 15,22; p.99: 13.
    SOLUTIONS here

    Homework 3, due Oct 1st 2010:
    9th edition: p.144: 1, 9, 13, 23; p.155: 1.
    OR 8th edition: p.142: 1, 9, 13, 23; p.151: 1.
    SOLUTIONS here

    Homework 4, due Oct 8th 2010:
    9th edition: p.163: 2, 17, 29, 32; p.171: 23; p.183: 17, 28.
    OR 8th edition:p.164: 2, 17, 29, 32; p.173: 23; p.184: 17, 28.
    SOLUTIONS here

    Homework 5, due WEDNESDAY Oct 20th 2010:
    9th edition: p.189: 1, 19, 21, 28; p.202: 5, 15, 16; p.216: 17.
    OR 8th edition:p.190: 1, 19, 21, 28; p.203: 5, 15, 16; p.214: 17.
    SOLUTIONS here

    Homework 6, due Friday October 29th 2010:
    9th edition: p.311: 14, 18, 26; p.320: 27a; p.328: 13, 25, 29, 30.
    OR 8th edition:p.312: 14, 18, 26; p.322: 27a; p.329: 7, 19, 23, 24.
    SOLUTIONS here

    Homework 7, due Friday November 5th 2010:
    9th edition: p.336: 1, 10; p.343: 25; p.350: 7, 13, 22, 29.
    OR 8th edition: p.337: 1, 10; p.344: 25; p.351: 7, 13, 22, 29.
    Note: p.351: 22b,c and 29 will not be graded.
    SOLUTIONS here and, for 6.4.1 and 6.4.10, here.

    Homework 8, due Friday November 12th 2010:
    9th edition:p.398: 15, 28, 29, 32, 33; p.409: 26, 27; p.428: 1.
    OR 8th edition: p.398: 15, 28, 29, 32, 33; p.410; 26, 27; p.428: 1.
    SOLUTIONS here and, for 7.8.1, here.

    Homework 9, due Friday November 19th 2010:
    9th edition: p.439: 1,3; p494: 2(a)-(c).
    OR 8th edition: p.439: 1,3; p492: 2(a)-(c).
    SOLUTIONS here and, for 7.9.3: undetermined vectors, variation of vectors.

    Homework 10, due Friday November 26th 2010:
    9th edition: p.494: 3, 4, 5; Page 506: 19.
    OR 8th edition: p.492: 3, 4, 5; Page 501: 17..
    SOLUTIONS here.

    Homework 11, due Wednesday Dec 1st 2010:
    9th edition: p.516: 5,6, 19, 27, 30.
    OR 8th edition: p.511: 5,6, 19, 26, 28.
    SOLUTIONS here.

    Documents

    (1) Applications with first-order equations: worked-out examples
    (2) Problems from problem session (27 Sept 2010): here
    (3) Reduction of order example
    (4) Method of undetermined coefficients table of trial expressions
    (5) Variation of parameters example
    (6) Forced Oscillations example
    (7) Summary of the method of partial fractions expansion (excerpt from Nagle, Saff, and Snider's Fundamentals of Differential Equations and Boundary Value Problems).
    (8) Discontinuous Forcing Functions example
    (9) SAME Discontinuous Forcing Functions example solved using convolution integrals. Includes details on integrating over step functions.
    (10) For enrichment only, NOT course material: Handout on computing the inverse Laplace transform via complex integration. The handout shows the basic steps and discusses a couple of the theorems used; it also contains citations for proofs and further reading.
    (11) Problems from problem session (3 Nov 2010): here. Please inform me immediately if you find any typos. UPDATED 13 Nov 2010 (one typo corrected).
    (12) Summary of linear systems in the plane - characterization of critical points and phase portraits (excerpt from Nagle, Saff, and Snider's Fundamentals of Differential Equations and Boundary Value Problems).
    (13) Examples of how to compute eigenvalues and eigenvectors of a matrix.
    (14) Problems from problem session (15 Nov 2010): here. Please note that these are the problems I had prepared in advance. At the problem session based on questions some slightly different problems were computed, with emphases placed on different concepts.

    Lecture Schedule (tentative)

    Week Topic Reading Suggested Problems Lecture Notes
    September 6 I. Introduction: what is a DE, order, linear and nonlinear, solution, general solution, particular solution.
    II. First Order Equations: solution of linear ODE, direction field    		 9th ed.:
    Chap. 1, 2.1.

    8th ed.:
    Chap. 1, 2.1.     		9th edition:
    p.15: 1(a), 3, 4, 8, 13, 15, 17, 18; p.24: 18, 20.
    p.39: 5, 11, 14, 21, 24, 32; p.99: 13.
    8th edition:
    p.15: 1(a), 3, 4, 8, 13, 15, 17, 18; p.24: 18, 20.
    p.39: 5, 11, 14, 21, 24, 32; p.99: 13.     		Lecture 2
    September 13 II. First Order Equations: existence and uniqueness, integrating factors, separable equations, symmetry, homogeneous equations, applications 9th ed.:
    2.2 - 2.6.

    8th ed.:
    2.2 - 2.6.     		9th edition:
    p.75: 3, 25, 27; p.47: 1, 6, 30, 34; p.59: 8, 9, 10, 16, 18, 32; p.88: 15, 20, 22, 24 , 28.
    8th edition:
    p.75: 3, 25, 27; p.47: 1, 6, 30, 34; p.59: 8, 9, 10, 16, 18, 32; p.88: 15, 20, 22, 24 , 28.     		Lecture 3
    Lecture 4
    Lecture 5
    September 20 III. Second order linear equations: linear operator, existence and uniqueness, linear independence, linear homogeneous equation, linear nonhomogeneous equation 9th ed.:
    3.1, 3.2.
    8th ed.:
    3.1, 3.2.     		9th edition:
    p.144: 1, 9, 13, 17, 23, 28; p.155: 1, 2, 46, 51.
    8th edition:
    p.142: 1, 9, 13, 17, 23, 28; p.151: 1, 2, 33, 38.     		Lecture 6
    Lecture 7
    Lecture 8
    September 27 III. Second order linear equations: Wronskians and linear independence (fundamental set of solutions), constant coefficient linear homogeneous equations (characteristic equation: real roots, double roots, complex roots), linear nonhomogeneous equation (method of undetermined coefficients when the homogeneous equations has constant coefficients) 9th ed.:
    3.2, 3,3, 3.4, 3.5.
    8th ed.:
    3.2, 3,3, 3.4, 3.5, 3.6.     		9th edition:
    p.163: 2, 7, 17, 25, 29, 32, 34; p.171: 1, 14, 23; p.183: 1, 8, 17, 28, 29.
    8th edition:
    p.164: 2, 7, 17, 25, 29, 32, 38; p.172: 1, 14, 23; p.184: 1, 8, 17, 28, 29.     		Lecture 9
    Lecture 10
    Lecture 11
    October 4 III. Second order linear equations: linear nonhomogeneous equation (method of undetermined coefficients when the homogeneous equations has constant coefficients), linear nonhomogeneous equation (method of variation of parameters), applications to electrical circuits and mechanical vibrations

    MIDTERM     		9th ed.:
    3.5, 3.6, 3.7.
    8th ed.:
    3.6, 3.7, 3.8.     		9th edition:
    p.189: 1, 5, 19, 21, 28, 29; p.202: 5, 15, 16, 19, 20, 30.
    8th edition:
    p.190: 1, 5, 19, 21, 28, 29; p.203: 5, 15, 16, 19, 20, 30.     		Lecture 12
    Lecture 13
    October 11 III. Second order linear equations: applications to electrical circuits and mechanical vibrations
    		 9th ed.:
    3.7, 3.8.
    8th ed.:
    3.8, 3.9.     		9th edition:
    p.215: 1, 5, 17.
    8th edition:
    p.214: 1, 5, 17.     		Lecture 14
    Lecture 15
    October 18 IV. Laplace Transform: definition and examples, solution of initial value problems, discontinuous functions 9th ed.:
    6.1, 6.2, 6.3.

    8th ed.:
    6.1, 6.2, 6.3.     		9th edition:
    p.311: 5, 6, 14, 18, 26, 27; p.320: 2, 11, 20, 24, 27(a,b), 28, 30, 37; p.328: 13, 25, 29, 30, 33, 34.
    8th edition:
    p.312: 5, 6, 14, 18, 26, 27; p.322: 2, 11, 20, 24, 27(a,b), 28, 30, 37; p.329: 7, 19, 23, 24, 27, 28.     		Lecture 16
    Lecture 17
    Lecture 18
    October 25 IV. Laplace Transform: discontinuous functions, impulse functions 9th ed.:
    6.4, 6.5.
    8th ed.:
    6.4, 6.5.    		 9th edition:
    p.336: 1, 10, 19; p.343: 1, 25.
    8th edition:
    p.337: 1, 10, 19; p.344: 1, 25.     		Lecture 19
    Lecture 20
    November 1 IV. Laplace Transform:impulse functions, convolutions

    V. Systems of first order linear equations: homogeneous case    		 9th ed.:
    6.5, 6.6, 7.5.
    8th ed.:
    6.5, 6.6, 7.5.    		 9th edition:
    p.350: 1, 7, 13, 21, 22, 29; p.398: 1, 15, 29, 32, 33.
    8th edition:
    p.351: 1, 7, 13, 21, 22, 29; p.398: 1, 15, 29, 32, 33.
    		 Lecture 21
    Lecture 22
    Lecture 23
    November 8 V. Systems of first order linear equations: homogeneous case, non-homogeneous case

    		9th ed.:
    7.6, 7.8, 7.9, 9.1.

    8th ed.:
    7.6, 7.8, 7.9, 9.1.    		 9th edition:
    p.409: 1, 26, 28; p.428: 1; p.439: 1, 3; p.494: 1(a-c), 17, 20, 21.
    8th edition:
    p.410: 1, 26, 28; p.428: 1; p.439: 1, 3; p.492: 1(a-c), 17, 20, 21.     		Lecture 24
    Lecture 25
    Lecture 26
    Example from Lecture 26
    November 15 V. Systems of first order linear equations: non-homogeneous case

    VI. Nonlinear Systems: introduction, example of simple pendulum

    MIDTERM     		9th ed.:
    7.9, 9.1, 9.2.
    8th ed.:
    7.9, 9.1, 9.2.    		 9th edition:
    p.506: 1, 3, 17, 21, 23.
    8th edition:
    p.501: 1, 3, 17, 21, 23.     		Lecture 27
    Lecture 28
    November 22 VI. Nonlinear Systems: introduction, example of simple pendulum. 9th ed.:
    9.3, 9.4, 9.5.
    8th ed.:
    9.3, 9.4, 9.5.    		 9th edition:
    p.516: 1-6, 19, 21, 22, 27.
    8th edition:
    p.511: 1-6, 17, 19, 20, 25.     		Lecture 29
    Lecture 30
    Lecture 31
    November 29 VI. Nonlinear Systems: critical points (type/stability), phase portraits, applications.

    VII. Catch Up? And/or Review?: may be used for lectures to catch-up on schedule otherwise for review    		 9th ed.:
    9.3, 9.4, 9.5.
    8th ed.:
    9.3, 9.4, 9.5.    		 . Lecture 32
    Lecture 33
    Lecture 34

    Other sections

    Section 101 (Bluman)
    Section 102 (Nec)
    Section 103 (Schoetzau)
    Section 105 (Dridi)

    Notes

    Jessica M. Conway / Department of Mathematics / University of British Columbia