Math 251H, Section 1: Ordinary and Partial Differential Equations (Fall 2017)

Instructor: Dr. Jessica M. Conway.
Lectures: MWRF 12:20 - 1:10pm, Huck Life Sciences Bldg 005.
Office hours: McAllister 332, Mondays 9:30-10:30am, Fridays 1:30-3:30, and by appointment.
Email: jmconway (at) psu (dot) edu
Phone: (814)863-9125
PSU offers Mathematics drop-in and online tutoring, ODEs/PDEs included!
Schedule and locations available at https://pennstatelearning.psu.edu/tutoring/mathematics.
Course Syllabus available HERE.

Text: Boyce and Diprima, Elementary differential equations and boundary value problems, 10th or 11th edition.
We will cover chapters 1-3, 4, 6-7, 9-10. Chapter 5 will be covered as independent study.
Alternatives to new hardcover: there's an as an e-text and also a loose leaf version, if not from the University Bookstore, then from Amazon or the publisher website (Wiley). The text can be rented from amazon.

Additional materials:
  • We'll use software to understand direction fields and phase planes, available at http://math.rice.edu/~dfield/dfpp.html.
  • MATLAB may be used for a few demonstration; if so, code will be provided. It is available for free as a WebApp at https://webapps.psu.edu/.
  • Web-based software will be used for demonstrations in sections 3.7/3.8. It will be provided when the time comes.
  • Dr. Wen Shen's Lecture Notes for Math 251/251H HERE. She is generously sharing them with us as an additional resource. There are LOTS of examples!
  • Vibrations web application: link.
  • Forced Vibrations web application: link.

    • ANNOUNCEMENTS:

  • (08/21/2017) Welcome to Math 251H!
  • (15/09/2017) "Vibrations" web application posted in "Additional Materials" section.
  • (25/09/2017) "Forced Vibrations" web application posted in "Additional Materials" section.

    • Exam Dates:

  • Midterm test 1: Thursday, October 5, 2017; 6-7:15pm, 362 Willard.
  • Midterm test 2: Monday, November 6, 2017; 6-7:15pm, 362 Willard.
  • Final exam: December 13, 2017; 12:20-2:10pm; 62 Willard.

    • Grading

    50 points (10%) from homework + 75 points (15%) from quizzes + 25 points (5%) from the independent study unit + 100 points (20%) from Midterm 1 + 100 points (20%) from Midterm 2 + 150 points (30%) from the Final Exam = 500 points (100%).

    Homework

    Due each week at the beginning of the Monday's class unless otherwise stated. For now, homework will only be checked for completion, but not graded.
    Warning: if you all start "phoning it in", homework will be partially graded.

    Late home work will NOT be accepted.

    Quizzes

  • 15-20 minutes, every Wednesday, based on the previous week's homework.
  • We'll drop the lowest two quizzes to calculate the final quiz grade.

    • Quiz SOLUTIONS:
    Quiz 1 solutions.
    Quiz 2 solutions.
    Quiz 3 solutions.
    Quiz 4 solutions.
    Quiz 5 solutions.
    Quiz 6 PRACTICE: quiz and solutions.
    Quiz 6 solutions.
    Quiz 7 solutions.
    Quiz 8 solutions.
    Quiz 9 solutions.
    Quiz 10 solutions.
    Take home Quiz 11 link. Due Monday Nov 13. Do not discuss with classmates or others. Solutions.
    Quiz 12 solutions.
    Quiz 13 solutions.

    Exams

    Old exams are available in a repository maintained by Dr. Zachary Tseng, here.

    Midterm 1 prep
  • Topics overview + Series solutions to ODEs, ISU problem sets 1-3 (Sections 5.1-5.3 of textbook).
  • 2016: midterm and solutions.
  • Your exam: midterm and solutions.


    • Midterm 2 prep
  • Topics overview.
  • 2016: midterm and solutions.
  • Your exam: midterm and solutions.


    • Final exam prep
  • Topics overview.
  • 2016: midterm and solutions.


    • Independent study unit

    On Series Solutions to ODEs (Chap. 5 of Boyce & DiPrima). I will not be covering this material in lecture. Evaluation will be based on problem sets which will be fully graded. The material is fair game on midterms and the final exam.

    Problem sets
    Problem set 1, due September 8, 2017. Review on Taylor series + material from section 5.1 of text. Solutions.
    Problem set 2, due September 22, 2017. Material from sections 5.2 and 5.3 of text. Solutions.
    Problem set 3, due September 29, 2017. Material from sections 5.2 and 5.3 of text. Solutions.
    Problem set 4, due October 20, 2017 OR December 8, 2017. Material from sections 5.4 of text. Solution to bonus problem that may help with Problem Set 5 (end of page 1/start of page 2...). Solutions.
    Problem set 5, due Nov 3, 2017 OR December 8, 2017. Material from sections 5.5 of text. Solutions.
    Problem set 6, due Nov 17, 2017 OR December 8, 2017. Material from sections 5.6 of text. Solutions.

    Documents

    (1) Applications with first-order equations: worked-out examples.
    (2) Additional reduction of order example.
    (3) Brief discussion of hyperbolic functions.
    (4) Method of undetermined coefficients table of trial expressions.
    (5) Three variation of parameters examples.
    (6) Videos showing the applications of resonance: bending & breaking a glass, bending & breaking a bridge.
    (7) Forced & free vibrations example.
    (8) Greek alphabet.
    (9) Refresher on polynomial long division: Khan Academy Video.
    (10) Seventh order, linear, homogeneous ODE general solution example.
    (11) 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.
    (12) Table of Laplace transforms: link. This table is the one that will be provided for quizzes and exams. It contains the same information as Table 6.2.1 in your textbook.
    (13) Summary of the method of partial fractions expansion (excerpt from Nagle, Saff, and Snider's Fundamentals of Differential Equations and Boundary Value Problems).
    (14) Discontinuous forcing functions example.
    (15) Same discontinuous forcing functions example using convolution.
    (16) Laplace transforms of periodic functions: derivation.
    (17) For interested students: Matlab code to graph the solution of the last initial value problem in Lecture 30.
    (18) Examples of eigenvalue & eigenvector calculations: link.
    (19) Heat conduction problem example, 10.6.10: link.

    Lecture Schedule (tentative; will be updated as semester proceeds)

    Week Topic Reading Homework (ANSWERS at back of book) Lecture Notes
    Aug 21 I. Introduction: what is a DE, order, linear and nonlinear, solutions, direction fields.
    II. First Order Equations: solution of linear ODE, separable equations, integrating factors.    		 Chap. 1, 2.1-2.2. 10th Edition:
    1.1: 5, 9, 13, 15-20, 21, 23
    1.3: 1, 3, 5, 7, 12, 17, 19
    2.1: 4, 11, 13, 16, 20, 24, 31, 33
    2.2: 7, 17, 19, 21, 24, 27, 30, 31, 33
    		11th Edition:
    1.1: 4, 6, 9, 11-16, 17, 19
    1.3: 1, 3, 4, 5, 9, 12, 14
    2.1: 4, 7, 9, 11, 12, 15, 21, 23
    2.2: 5, 15, 16, 17, 20, 21, 25, 26, 28
    		Lecture 1
    Lecture 2
    Lecture 3
    Lecture 4
    Aug 28 II. First Order Equations: integrating factors, modeling and applications, linear vs nonlinear equations / existence & uniqueness, autonomous equations & population dynamics 2.3 - 2.5, 2.8. 2.3: 5, 9, 13, 27
    2.4: 5, 11, 13, 25, 27, 29
    2.5: 5, 7, 13, 15, 17, 21, 23
    Enrichment problems, not required:
    The Brachistochrone: 2.3.32
    More bifurcations: 2.5.25, 27     		2.3: 3, 7, 9, 21
    2.4: 3, 8, 9, 21, 23, 24
    2.5: 4, 5, 9, 15, 17, 20, 22
    Enrichment problems, not required:
    The Brachistochrone: 2.3.24
    More bifurcations: 2.5.24, 26     		Lecture 5
    Lecture 6
    Lecture 7
    Lecture 8
    Sep 4 II. First Order Equations: Exact equations & integrating factors.
    III. Second order linear equations: Homogeneous equations, the Wronskian.     		2.6, 3.1 - 3.2 2.6: 5, 6, 9, 15, 19, 27, 28, 32
    3.1: 5, 8, 15, 19, 21, 23, 26
    3.2: 9, 13, 15
    Enrichment problems, not required:
    Exact equations: 3.2.41,45     		2.6: 4, 5, 6, 11, 15, 20, 21, 22
    3.1: 4, 6, 11, 14, 16, 17, 19
    3.2: 7, 10, 12
    Enrichment problems, not required:
    Exact equations: 3.2.31, 34     		Lecture 9
    Lecture 10
    Sep 11 III. Second order linear equations: The Wronskian, characteristic equations, real, complex, an repeated roots, reduction of order. 3.2 - 3.4 3.2: 17, 28, 29
    3.3: 11, 17, 19, 22, 25
    3.4: 7, 11, 14, 17, 23, 27, 37, 38
    Enrichment problems, not required:
    Euler equations: 3.4.41, 43
    Try also the substitution y(t)=tr, solve for r, then write down the general solution. This is the usual sub for Euler eqs.     		3.2: 14, 22, 23
    3.3: 8, 12, 13, 15, 18
    3.4: 6, 9, 11, 13, 18, 21, 28, 29
    Enrichment problems, not required:
    Euler equations: 3.4.32, 33
    Try also the substitution y(t)=tr, solve for r, then write down the general solution. This is the usual sub for Euler eqs.     		Lecture 12
    Lecture 13
    Lecture 14
    Lecture 15
    Sep 18 III. Second order linear equations: vibrations, nonhomogeneous equations, method of undetermined coefficients, variation of parameters. 3.5-3.7 3.5: 5, 6, 11, 13, 17, 19, 23a, 27a, 29, 31
    3.6: 1, 7, 17, 29
    3.7: 1, 7, 11, 13, 19, 21, 24
    Note: 3.5.13 involves hyperbolic functions. If these are new to you, see the note on hyperbolic functions in 'Documents'.     		3.5: 4, 5, 8, 10, 13, 14, 17a, 20a, 22, 24
    3.6: 1, 6, 13, 24
    3.7: 1, 4, 6, 8, 13, 15, 17
    Note: 3.5.10 involves hyperbolic functions. If these are new to you, see the note on hyperbolic functions in 'Documents'.     		Lecture 16
    Lecture 17
    Lecture 18
    Lecture 19
    Sep 25 III. Second order linear equations: forced vibrations, application: mechanical and electrical vibrations.
    IV. Higher order linear equations: Existence & uniqueness of solutions, solving constant-coefficient higher-order linear equations.
    		 3.8, 4.1, 4.2 3.7: 12
    3.8: 6, 8, 13, 15, 16, 18, 19, 24
    4.1: 5, 9, 15, 19, 23, 27 (don't have to show #26)
    4.2: 5, 7, 24, 35, 39
    		3.7: 7
    3.8: 4, 5, 9, 11, 12, 13, 14, 19
    4.1: 3, 7, 10, 14, 17, 20 (don't have to show #19)
    4.2: 4, 5, 17, 25, 29
    		Lecture 20
    Lecture 21
    Lecture 22
    Lecture 23
    Oct 2 VI. Laplace Transform: definition, initial value problems.

    MIDTERM I     		6.1, 6.2 6.1: 3, 7, 19, 23, 27
    6.2: 1, 9, 11, 17, 22, 27, 29
    Enrichment problems, not required:
    Gamma functions: 6.1.30, 31     		6.1: 3, 6, 15, 17, 21
    6.2: 1, 7, 8, 13, 16, 19, 21
    Enrichment problems, not required:
    Gamma functions: 6.1.23, 24     		Lecture 24
    Oct 9 VI. Laplace Transform: step functions, IVPs with discontinuous functions, impulse functions. 6.3 - 6.5 6.3: 3, 12, 15, 17, 21, 33, 34, 35
    6.4: 10, 13, 15, 19
    6.5: 7, 11, 13, 18, 23
    		6.3: 2, 8, 10, 12, 15, 23, 24, 25
    6.4: 6, 8, 10, 15
    6.5: 5, 7, 9, 14, 17
    		Lecture 27
    Lecture 28
    Lecture 29
    Lecture 30
    Oct 16 VI. Laplace Transform: Convolution integrals.
    VII. Systems of first order linear equations: 2x2 matrices, linear systems of differential equations.     		6.6, 7.1-7.5 6.6: 1, 7, 9, 15, 18, 22
    7.1: 1, 11, 18, 23
    7.2: 1, 8, 23
    7.3: 13, 17, 20, 29
    7.5: 15, 27, 33
    Enrichment problems, not required:
    The Tautochrone: 6.6.29     		6.6: 1, 6, 8, 12, 14, 17
    7.1: 1, 9, 15, 20
    7.2: 1, 7, 17
    7.3: 11, 15, 17, 24
    7.5: 10, 19, 25
    Enrichment problems, not required:
    The Tautochrone: 6.6.22     		Lecture 31
    Lecture 32
    Lecture 33
    Lecture 34
    Oct 23 VII. Systems of first order linear equations: Linear systems of differential equations. 7.5 - 7.9 7.5: 3, 19, 20, 25
    7.6: 5, 13, 21, 26, 28
    7.8: 3, 7, 13, 16
    7.9: 1, 3, 7, 13
    Enrichment problems, not required:
    Variation of parameters: 7.9.5, 15
    Laplace transforms: 7.9.18
    		7.5: 3, 13, 14, 18
    7.6: 3, 11, 16, 21, 23
    7.8: 3, 6, 11, 14
    7.9: 1, 2, 5, 9
    Enrichment problems, not required:
    Variation of parameters: 7.9.4, 11
    Laplace transforms: 7.9.17
    		Lecture 35
    Lecture 36
    Lecture 37
    Lecture 38
    Oct 30 VII. Systems of first order linear equations: Inhomogeneous systems.
    IX. Nonlinear Systems: Nonlinear systems of equations and stability.     		9.1 - 9.4 DUE MONDAY NOV 13; TAKE HOME Quiz 11 DUE NOV 13
    9.1: 7 (omit (d)), 13, 18, 20, 21
    9.2: 7, 13, 15, 17, 24 (7-15 use pplane, don't need to turn in graph)
    9.3: 13, 15, 17, 23
    9.4: 5, 12, 17
    Enrichment problems, not required:
    Bifurcations: 9.4.13, 15     		DUE MONDAY NOV 13; QUIZ 11 is TAKE-HOME
    9.1: 7 (omit (d)), 11, 15, 17, 18
    9.2: 6, 10, 12, 14, 20 (6-12 use pplane, don't need to turn in graph)
    9.3: 10, 12, 14, 20
    9.4: 3, 10, 15
    Enrichment problems, not required:
    Bifurcations: 9.4.11, 13     		Lecture 39
    Example from Lecture 39.
    Lecture 40
    Lecture 41
    Lecture 42
    Nov 6 IX. Nonlinear Systems: Nonlinear systems of equations and stability.
    X. PDEs and Fourier Series: Two-point boundary value problem, Fourier Series and convergence

    MIDTERM II MONDAY     		9.3-9.4, 10.1 - 10.3 Lecture 43
    Lecture 44
    Lecture 45
    Nov 13 X. PDEs and Fourier Series: even & odd functions, separation of variables, heat conduction. 10.4 - 10.6 10.1: 16, 19
    10.2: 3, 7, 11, 21
    10.3: 3, 7, 14, 17
    10.4: 5, 6, 11, 19, 37
    10.5: 3, 5, 12, 17 (omit (d)), 18
    10.6: 9, 13, 21, 23
    Sample Mathematica script to help you plot partial sums and errors.
    Enrichment problems, not required:
    10.1: 11, 13, 21
    10.4: 35, 36     		10.1: 16, 19
    10.2: 3, 7, 11, 21
    10.3: 3, 7, 14, 17
    10.4: 5, 6, 11, 19, 37
    10.5: 3, 5, 12, 17 (omit (d)), 18
    10.6: 9, 13, 21, 23
    Sample Mathematica script to help you plot partial sums and errors.
    Enrichment problems, not required:
    10.1: 11, 13, 21
    10.4: 35, 36     		Lecture 46
    Lectures 45/46 MATLAB demo script.
    Lecture 47
    Lecture 48
    Lecture 48 MATLAB demo script.
    Lecture 49
    Nov 20 THANKSGIVING BREAK
    Nov 27 X. PDEs and Fourier Series: Laplace equation. 10.8 10.8: 3, 5, 7, 9, 11, 14, 16 10.8: 3, 5, 7, 9, 11, 14, 16 Lecture 50
    Lecture 51
    Lecture 52
    Dec 4 X. PDEs and Fourier Series: Wave equations, d'Alembert solution of the wave equation in an infinite string.

    FINAL EXAM REVIEW    		 10.7 10.7: 7 (omit (d)), 9, 11 (omit (d)), 15, 16, 21
    Enrichment: 10.7.22, 23     		10.7: 7 (omit (d)), 9, 11 (omit (d)), 15, 16, 21
    Enrichment: 10.7.22, 23     		Lecture 53
    Lecture 53-55 MATLAB demo script.
    Lecture 54
    Lecture 55
    Jessica M. Conway / Department of Mathematics / Pennsylvania State University