Math 246, Ordinary Differential Equations

Textbooks

  • C. David Levermore, Ordinary Differential Equations,see
  • https://courses.math.umd.edu/math246/NODE/1718S/main.html
  • login with your University of MD account
  • B.Hunt et. al., Differential Equations with MATLAB, 3d ed.

    Contact

  • Room: Math 3317. Phone: (301)405-5152. email: mvy@math.umd.edu
  • Office hours: TTh 10:45-11:45 and by appointment.

    Lectures

  • Armory 0126, TuTh 12:30-1:45

    Discussion Sections

  • Discussions on Wednesday
  • Sections
  • 0311 in math 0105, 0312 in math 0307, at 8 am
  • Sections
  • 0321 in math 0105, 0322 in math 0307, 0323 in math 0403 at 9 am
  • Sections
  • 0331 in math 0106, 0332 in math B0427 at 10 am
  • Sections
  • 0341 in math 0403, 0342 in math 0304 at 11 am -->

    Tutoring

  • Look at: www.math.umd.edu. Go to undergraduate/resourses.

    Grading

  • There will be a total of 700 points available in the course, allocated as follows.
  • Three exams, each worth 100 points, see tentative dates below.
  • Several quizzes, each worth 25 points, 4 best count toward the total of 100 points, see tentative dates below.
  • Six MATLAB assignments, each worth 20 points. 5 best count toward the total of 100 points. See below due dates for the submission .
  • Cumulative final exam worth 200 points.

    Missed exams, quizzes

  • Please note that for a missed exam/quiz a written explanation from a doctor will need to be provided. With the proper explanation 1/2 of your score for the Final Exam will substitute the score for the missed exam.
  • For a missed quiz with the proper explanation 1/4 of your score for the next exam will substitute the score for the missed quiz.
  • You may appeal the score you receive on an exam/quiz by submitting your exam/quiz and a note stating which problems you wish to have regraded. For quizzes you can do it within one week of the quiz. For exams 1 and 2 you can do it within two weeks of the exam. For exam 3 by the time of the Final exam Review.
  • Tentative exam dates. Exam 1- Feb.27. Exam 2 - March 29. Exam 3 - May 3.
  • ATTENTION. FINAL EXAM.May 12, 1:30-3:30. Please arrive at 1:20. Bring photo ID.

    Exams

  • Exam 1. February 27. Material: Lvrmr, Ch. 1: Sections 1-8. See Training
  • Exam 2. Tentative date March 29. Material: Lvrmr, Ch. 2 Sections 1- 8.
  • Recommended to review for exam 2. (a) Integration technique. (b) The method of integrating factor for 1-st order linear ODE. (c) Wronskian (Section 2). (d) Constant coefficients (Section 6). Problems recommended for review: 10-12, 15,16, 25 ; Review internal link in Section 6 "summary of methods". (e) Variable coefficients (Section 7). Problems recommended for review: 2,9,20. (f) Applications (Section 8). Problems recommended for review: 2,5-10,14,26,28-30. Review types of motion: harmonic, underdamped, overdamped, critically damped, resonance, steady state.
  • Exam 3.Tentative date May 3. Material: Lvrmr, Ch.3 Sections 1-6 and Section 8.
  • To prepare for exam 3 use problems from Section 4 Ex. 2-15, Section 5 Ex.7-26, Section 6 Ex.9-23. In each problem you should not only answer the questions of that problem, but answer all set of questions similar to questions of Quiz 6 : eigenvalues, eigenvectors, matrix exponential, solution of IVP, phase portrait, sketch solution of IVP in the phase plane. Choose yourself initial conditions, for example (1,0) or (0,1) or (1,1) or (1,-1) etc. In Section 8 find stationary solutions in exercises 1-6. In exercises 7,10,12-14 in addition to questions of the book, sketch phase portraits near each critical point.
  • ATTENTION. FINAL EXAM. May 12, 1:30-3:30. Please arrive at 1:20. Bring photo ID.

    Homeworks and Quizzes

  • Matlab. Problem Set A. Problems 5,8,9,10. Due Jan 31.
  • Study Chapters 1-4 of DE with Matlab. See sample solution in Section 4.4. You work in groups of 3 on all Matlab assignments. Matlab HW are accepted only in printed form signed by all members of the team. Not accepted on the web.
  • Quiz 1.Feb. 6 . Linear, Separable equations.Chapter 1, Sections 2 and 3.Training: Problems 4,10,15,19 - Section2, Problems 7,10,15,19 Section 3.
  • Recommended HW problems Chapter 1.
  • Section 2 Recommended HW problems:1-15.
  • Section 3 Recommended HW problems:1-20.
  • Section 4 Recommended HW problems:1-12.
  • Section 5 Recommended HW problems:1-23.
  • Section 6 Recommended HW problems:1-17.
  • Section 7 Recommended HW problems:1-10.
  • Section 8 Recommended HW problems:1-21.
  • PROBLEM SET B. Due Feb. 14. Problems 3,13,19,21. Study Section 5 "Graphical methods" of Lectures. Study Chapters 5,7 of DE with Matlab. It is recommended to look at sample solutions at the end of the Matlab Book.
  • Answer all questions. Detailed explanations are recommended. In problems 13 and 19 in addition to the book's questions do also the following. Use "solve" and/or "fzero" commands to find equilibrium (stationary) solutions. Then use "plot" command to plot the graph of f(y) and check that zeros of the graph coincide with equilibria you found. Then plot (by hand) the phase-line portraits, and determine type (repelling, attracting etc) of equilibrium solutions based on the phase-line portraits. Use that information when you analyze the vector fields.
  • Quiz 2. Feb. 15. Applications and first sections in numerical methods. Study sections 5 and 6 of the book and examples from lectures. Also study Euler method and errors. Study sections 7.2.1, 7.3.1, 7.3.2.
  • Exam 1. Feb. 27. Topics see above in exams.
  • PROBLEM SET C. Due March 7 . Problems 1,10,14. Study Section 7 of Chapter 1: Numerical methods. Also study Chapters 3-8 of DE with Matlab. See sample solution of one of the problems at the end of the Matlab Book. Also see an example example. This example will work if you include in the same directory the function M-file myeuler myeuler. See Section 8.2.1 of Matlab book.
  • Quiz 3. Numerical methods and first sections of high order ODE. March 6 . For numerical methods study Section 7 of Chapter 1 and examples from lectures. Also study Chapters 3-8 of DE with Matlab. For high order ODE study Chapter II Sections 1-4.
  • Recommended HW problems, Chapter II.
  • Section 1. 1-10.
  • Section 2. 1-20.
  • Section 3. 1-10.
  • Section 4. 1-22.
  • Section 5. 13-19.
  • Section 6. 1-27.
  • Section 7. 1-14.
  • Section 8. 1-23.
  • Section 9. 1-20.
  • Problem set D. Due March 14. Problems 3,4,5.
  • COMMENTS to Problem set D.
  • In problem 3 plot the graphs of the linear approximation and of the actual pendulum . One can estimate the period T of nonlinear oscillations based on the graphs . Notice that the period equals twice the distance between two consecutive moments x1 such that y1(x1) = 0 . In order to find such moments you can plot graphs using the option ``axis''. For example
  • plot(x1,y1(:,1))
  • axis([1.56 \ 1.58 \ -0.001 \ 0.001])
  • Next in order to find a root more accurately you can use the ``zoom'' feature on the ``Figure'' window of Matlab.
  • Remark. We use plot(x,y(:,1)) not plot(x,y), because when we use plot(x,y), matlab plots not only y(t), but also an extra graph of velocity dy/dt, which is not needed here.
  • If you want ode45 to do more precise calculations you can use ``Options'' described in Section 7.3 of the Matlab book.
  • In problem 4 you can try to increase accuracy when the initial speed equals 2. In that case when time is large Matlab produces wrong graphs. First do your computations with the defaut accuracy, then increase it consecutively. Explain the difference between graphs and why eventually the graphs become wrong.
  • Solve problem 5 using the following values of damping coefficient : b = .5, 1, 2 . In this problem you can use Simulink or you can use function m-files. For example can use the following function m-file for the linear model: function ode = F(t,y,unused,b)
  • ode = [y(2); -b*y(2)-y(1)];
  • and call it for example Flinear.m
  • After that file is saved in the same directory as the main m-file you can use in the main m-file :
  • for b = [.5 ,1 ,2 ]
  • [t,y]= ode45('Flinear.m', [0 20], [0 4],[ ],b);
  • plot(t;y(:,1))
  • end
  • Overall there are several possibilities for bonus in that Project.
  • Quiz 4. March 15 . Material: Chapter 2. Sections 1-6,8.
  • Exam 2. March 29 . Topics see above in exams.
  • Problem Set E. Due April 11. Problems 10,12,13(a-c). Answer all questions.
  • Quiz 5. April 10 . Material: Chapter 2. Section 9. Ex. 1-20. Table of Laplace transforms, see "Internal links".
  • Systems. Chapter 3. Recommended HW problems.
  • Section 1. Recommended HW problems:1-10.
  • Section 2. Recommended HW problems:1-13.
  • Section 3. Recommended HW problems:1-7, 14-17, 20-22.
  • Section 4. Recommended HW problems:2-15,18.
  • Section 5. Recommended HW problems:1-22.
  • Section 6. Recommended HW problems:1-23. In all problems determine type and stability.
  • Section 7. Recommended HW problems:1-23.
  • Section 8. Recommended HW problems:1-18.
  • Section 9. Recommended HW problems:1-14.
  • Quiz 6. April 24 . Chapter 3. Sections 4-6.
  • To prepare for quiz 6 use problems from Section 4 Ex. 2-15, Section 5 Ex.7-26, Section 6 Ex.9-23. In each problem you should not only answer the questions of that problem, but answer all set of questions : eigenvalues, eigenvectors, matrix exponential, solution of IVP, phase portrait, sketch solution of IVP in the phase plane. Choose yourself initial conditions, for example (1,0) or (0,1) or (1,1) or (1,-1) etc.
  • Problem set F. Due May 2 . Problems 1, 5. When answering question 5(f) classify the type and stability of the critical points (0,0) and (pi,0).
  • COMMENTS to Problem set F.
  • Problem 1.
  • You can write general solutions by hand. When writing use constants c1 and c2.
  • When answering question (b) for the first equation answer an additional question. Let matlab solve initial value problem with initial conditions x(0)=a, y(0) = b . Then matlab expresses solution using constants a and b. Note that you did the same using constants c1 and c2. Find relation between constants (a,b) and constants (c1,c2).
  • Problem 5.
  • Question (c) is theoretical, just differentiate by hand. You can express E either as a function of theta(t)( which is a solution of the second order equation), or as a function of x(t), y(t) , which are solutions of the respective system. When evaluating dE/dt use the Chain Rule and after that use that theta (or x,y) satisfy given differential equation. Do the same when evaluating dE/dt in question (g).
  • In order to find b0 in question (f) do several approximations. First you define inline function, corresponding to our system, call it for example g. Then you can try something like
  • for b = 3:0.1:3.5
  • [t, xd] = ode45(g, [0 15], [0 b]);
  • plot (xd(:,1), xd(:,2))
  • and the graph shows where different trajectories go. You see where they diverge and based on that narrow the range of b. After you repeat that procedure several times you can find the required b0 with good precision.
  • Test 3. May 3. Topics see above in exams.
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