Math 251H, Section 1: Ordinary and Partial Differential Equations (Spring 2016)

Instructor: Dr. Jessica M. Conway.
Lectures: MTWF 10:10-11am, 271 Willard.
Office hours: McAllister 332, Mondays 3:30-5:30pm, Wednesdays 11:15-12:15 + 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 edition.
We will cover chapters 1-3, 4, 6-7, 9-10.
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/.
  • 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:

  • (11/01/2016) Welcome to Math 251H!
  • (22/01/2016) Midterm locations announced: 104 Thomas for both.
  • (22/01/2016) Midterm locations announced: 104 Thomas for both.
  • (08/02/2016) "Vibrations" web application posted in "Additional Materials" section.
  • (16/02/2016) "Forced Vibrations" web application posted in "Additional Materials" section.
  • (17/02/2016) Midterm 1 review materials posted.
  • (18/02/2016) Final exam date announced: May 4, 2016, 2:30-4:20pm. Location TBA.
  • (25/03/2016) Final exam location announced: 073 Willard.

    • Exam Dates:

  • Midterm test 1: Thursday, February 25, 2016; 6:30-7:45pm, 104 Thomas.
  • Midterm test 2: Monday, April 4, 2016; 6:30-7:45pm, 104 Thomas.
  • Final exam: Wednesday, May 4, 2016; 2:30-4:20pm, 073 Willard.

    • Grading

    50 points (10%) from homework + 75 points (15%) from quizzes + 25 points (5%) from the project + 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 Tuesday'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 Note: The original version of the quiz was far too long, but the solutions have more pieces worked out. And since extra examples can help some of us learn, original quiz solutions: link.
  • Quiz #5 solutions
  • Quiz #6 solutions
  • Quiz #7 solutions
  • Quiz #8 solutions
  • Quiz #9 solutions
  • Quiz #10 solutions
  • Quiz #11 solutions
  • Quiz #12 solutions
  • Quiz #13 solutions. ALSO: Quiz #13 solutions with corrected typo & discussion of the consequences of the changing initial condition, link.
  • Quiz #14 solutions

    • Exams

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

    Midterm 1
  • Topics overview.
  • Review problems: 1.1 12, 14; 2.1. 18, 22, 30; 2.2. 20, 26, 28; 2.3. 4, 14, 16; 2.4. 6, 14; 2.5. 10, 18, 28; 2.6. 8, 10, 26, 28; 3.1. 8, 14, 20; 3.2. 12, 14, 26, 30; 3.3. 16, 22; 3.4. 14, 16, 24; 3.5. 10, 20; 3.6. 16, 18; 3.7. 4, 10, 18, 26; 3.8. 4, 10, 18 (+ "Forced vibrations" example); 4.1. 6, 10; 4.2. 8, 18, 22, 34.
  • Midterm 1 and solutions.

    Midterm 2
  • Topics overview.
  • Review problems: 6.1. 18, 24; 6.2. 10, 20, 22, 37a; 6.3. 6, 14, 18, 24, 36; 6.4. 10; 6.5. 10, 12, 20; 6.6. 6, 10, 20, 24; 7.1. 2, 12, 22; 7.3. 6, 14, 20; 7.5. 6, 16, 26, 30, 32; 7.6. 6, 10, 20, 26, 28; 7.8. 4, 8, 10; 7.9. 6, 8.
  • Midterm 2 and solutions.

    Final exam
  • Topics overview.
  • Review problems (for remaining chapters): 9.1. 8, 10, 12; 9.2. 4, 18; 9.3. 14, 18; 9.4. 6, 8 (plot approximate trajectories, and don't forget the Chap. 9 quiz!); 10.1. 8, 10, 14, 18; 10.2. 10, 18; 10.3. 6 (don't forget about how Fourier series convergence works at jump discontinuities!); 10.4. 2, 12, 16, 17, 20; 10.5. 4, 8, 18; 10.6. 11, 12, 14, 22; 10.7. 4, 8, 10; 10.8. 6, 8, 10, 12.
  • Note: When it comes to plotting in Chap. 10, of course you're not expected to be able to plot things that you need a computer for! That's why we professors like initial conditions that exploit orthoginality so much...

    Projects

  • Due April 29, 2016.
  • You may work alone or with at most one partner.
  • Guidelines for project presentation: link.
  • List of projects: link.
    Note: No more than two groups per project, on a first-come, first-served basis. The list will be updated as projects are selected so that you can see what's available.

    • Available projects:
    All projects come from Nagle, Saff, & Snyder's Fundamentals of Differential Equations, 8th Edition.
    1. Oil spill in a canal: link.
    2. Differential equations in clinical medicine: link.
    3. Utility function and risk aversion: link (description starts near the bottom of the first page).
    4. Dynamics of HIV infection : link.
    5. Aquaculture: link.
    6. Market equilibrium: stability and time paths: link.
    7. Period doubling and chaos (requires Euler's method and RK4, i.e., some numerical solving): link (description starts partway down the first page).
    8. The simple pendulum: link.
    9. Spread of staph infection in hospitals: link.
    10. Things that bob (requires some numerical solving): link (description starts near the bottom of the first page).
    11. Hamiltonian systems: link (description starts near the bottom of the first page).
    12. Cleaning up the Great Lakes: link (description starts halfway down the first page).
    13. A growth model for phytoplankton: link (description starts halfway down the first page).
    14. Transverse vibrations of a beam: link.
    15. Higher-order difference equations: link (description starts near the bottom of the first page).
    16. Frequency-response modeling: link (description starts halfway down the first page).
    17. Spherically Symmetric Solutions to Schrödinger’s Equation for the Hydrogen Atom (series solutions of ODEs): link.

    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) Refresher on polynomial long division: Khan Academy Video.
    (9) Seventh order, linear, homogeneous ODE general solution example.
    (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) 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.
    (12) Summary of the method of partial fractions expansion (excerpt from Nagle, Saff, and Snider's Fundamentals of Differential Equations and Boundary Value Problems).
    (13) Discontinuous forcing functions example.
    (14) Same discontinuous forcing functions example using convolution.
    (15) Examples of eigenvalue & eigenvector calculations: link.
    (16) Heat conduction problem example, 10.6.10: link.
    (17) For interested students: MATLAB code used to generate figures for heat conduction example, and numerical answer to part (d), HeatConductionExampleFigs.m and HeatConductionForNLsolve.m.

    Lecture Schedule (tentative)

    Week Topic Reading Homework (ANSWERS at back of book) Lecture Notes
    January 11 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. 1.1: 5, 9, 13, 15-20, 21, 23
    1.3: 1, 3, 5, 7, 11, 17, 19
    2.1: 7, 11, 15, 17, 19, 23, 31, 33
    2.2: 7, 17, 19, 21, 25, 27, 30, 31, 33
    		Lecture 1
    Lecture 2
    Lecture 3
    Lecture 4 and bonus examples
    January 19 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, 29
    2.4: 5, 11, 13, 25, 27, 29
    2.5: 5, 7, 13, 15, 17, 19, 21, 23
    Enrichment problems, not required:
    The Brachistochrone: 2.3.32
    More bifurcations: 2.5.25, 27     		Lecture 5
    Lecture 6
    Lecture 7
    Bead on a hoop: setup
    January 25 II. First Order Equations: Exact equations & integrating factors, homogeneous equations 2.6, 3.1 - 3.2 2.6: 7, 11, 15, 21, 27, 29
    3.1: 5, 7, 15, 19, 21,23, 25
    3.2: 9, 13, 15
    Enrichment problems, not required:
    Exact equations: 3.2.41, 43
    		Lecture 8
    Lecture 9
    Lecture 10
    Lecture 11
    February 1 III. Second order linear equations: The Wronskian, characteristic equations, real, complex, an repeated roots, reduction of order. 3.2 - 3.4 3.2: 15, 17, 28, 31
    3.3: 9, 17, 19, 21, 25
    3.4: 9, 11, 13, 15, 23, 27, 37, 38
    Enrichment problems, not required:
    Euler equations: 3.4.34, 35
    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
    February 8 III. Second order linear equations: vibrations, nonhomogeneous equations, method of undetermined coefficients, variation of parameters. 3.5-3.7 3.5: 9, 11, 13, 14, 17, 19, 25a, 27a, 29, 31
    3.6: 3, 7, 19, 29
    3.7: 3, 7, 11, 17, 20, 24
    Note: 3.5.13, 14 involve hyperbolic functions. If these are new to you, see the note on hyperbolic functions in 'Documents'.     		Lecture 16
    Lecture 17
    Lecture 18
    Lecture 19
    February 15 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: 8
    3.8: 9, 13, 15, 17, 18, 19, 24
    4.1: 5, 9, 15, 19, 23, 27 (don't have to show #26)
    4.2: 5, 9, 23, 35, 39
    Enrichment problems, not required:
    4.1.20, 4.2.41
    Method of undetermined coefficients: 4.3.17
    Variation of parameters: 4.4.13     		Lecture 20
    Lecture 21
    Lecture 22
    Lecture 23
    February 22 VI. Laplace Transform: definition, initial value problems.

    MIDTERM I     		6.1, 6.2 6.1: 3, 9, 17, 23, 27
    6.2: 7, 9, 19, 21, 23, 27, 29
    Enrichment problems, not required:
    Gamma functions: 6.1.30, 31     		Lecture 24
    Lecture 25
    Lecture 26
    February 29 VI. Laplace Transform: step functions, IVPs with discontinuous functions, impulse functions. 6.3 - 6.5 6.3: 5, 11, 15, 17, 23, 34, 35
    6.4: 9, 13, 15, 19
    6.5: 9, 11, 13, 18, 23
    		Lecture 27
    Lecture 28
    Lecture 29
    Lecture 30
    March 7 SPRING BREAK
    March 14 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, 11, 17, 19, 23
    7.1: 1, 7, 18, 23
    7.2: 1, 9, 23
    7.3: 13, 17, 21, 29
    7.5: 5, 15, 27, 33
    Enrichment problems, not required:
    The Tautochrone: 6.6.29     		Lecture 31
    Lecture 32
    Lecture 33
    Lecture 34
    March 21 VII. Systems of first order linear equations: Linear systems of differential equations. 7.6 - 7.9 7.5: 19, 20, 25
    7.6: 5, 15, 21, 26, 28
    7.8: 3, 9, 13, 16
    7.9: 3, 5, 7, 13
    Enrichment problems, not required:
    Variation of parameters: 7.9.15
    Laplace transforms: 7.9.18
    		Lecture 36
    Lecture 37
    Lecture 38
    Example from Lecture 38.
    March 28 IX. Nonlinear Systems: Nonlinear systems of equations and stability. 9.1 - 9.4 9.1: 9, 11, 13, 16, 18, 20, 21
    9.2: 3, 11, 13, 15, 17, 24
    9.3: 3, 13, 15, 17, 23
    9.4: 3, 5, 12, 17
    Enrichment problems, not required:
    Bifurcations: 9.4.13, 15     		Lecture 39
    Lecture 40
    Lecture 41
    Lecture 42
    April 4 X. PDEs and Fourier Series: Two-point boundary value problem, Fourier Series and convergence

    MIDTERM II     		10.1 - 10.3 10.1: 16, 19
    10.2: 3, 7, 11, 15, 21, 25
    10.3: 3, 7, 14, 17
    Sample Mathematica script to help you plot partial sums and errors.
    Enrichment problems, not required:
    10.1: 11, 13, 21     		Lecture 43
    Lecture 44
    Lecture 45
    Lectures 44/45 MATLAB demo script.
    April 11 X. PDEs and Fourier Series: even & odd functions, separation of variables, heat conduction. 10.4 - 10.6 10.4: 1, 2, 3, 4, 11, 19, 21, 23, 35, 37
    10.5: 3, 5, 7, 12, 17 (omit (d)), 18
    10.6: 7, 9, 13, 21, 23     		Lecture 46
    Lecture 47
    Lecture 47 MATLAB demo script.
    Lecture 48
    Lecture 49
    April 18 X. PDEs and Fourier Series: Wave equation, Laplace equation. 10.7 - 10.8 10.7: 7 (omit (d)), 9, 11, 15, 21, 22, 23
    10.8: 3, 5, 7, 9, 11, 14, 16     		Lecture 50
    Lecture 51
    Lecture 52
    Lecture 53
    April 25 X. PDEs and Fourier Series: More wave equations, d'Alembert solution of the wave equation in an infinite string.

    FINAL EXAM REVIEW    		 10.8 Lecture 54
    Lecture 55
    Lecture 54 & 55 MATLAB demo script.
    Jessica M. Conway / Department of Mathematics / Pennsylvania State University