MAP 2302: Previous Assignments

MAP 2302 Honors Differential Equations

Previous Assignments

Happy New Year!
The required textbook and the recommended (optional) Student Solution Guide are already in the bookstore. Buy your copy and install the software from the book to your home computer.
Office hours: MW 3:30-6, F 10-11. By appointment only: Mon 6-7, Fri 3-5.
Read the Note to the Student in the textbook (page xiii).
Install DETools from the CD in the back pocket of the book to your home computer (open the file README on the CD and follow the instructions). Open FirstOrderExamples (the last tool in the first column) and play with the first four examples: y' = -y², y' = cos(t), y' = y(1-y), y' = -2ty². Drag the starting point with your mouse to obtain various solutions to each differential equation. Follow the specific instructions in Lab 0 and submit your work on the due date for Lab 0 (see below).
Note: If you cannot get the book and the CD-ROM by the due date, complete the assignment with the first 4 examples in the IDE Solution Tool (IDE Lab 1b) instead (the instructions in Lab 0 will not match these examples, but do something similar and meaningful).
Read the syllabus for policies. Ask questions next class.
Review the guidelines for Success in Mathematics.
Browse the rest of the course resources.
Read Sections 1.1 and 1.2.
1.1: 2-6, 10-17, 19-21. Submit: 4, 14, 16, 17, 20.
Wed 1/10 (next class): Lab 0: Ready, Set, Go.
Tue 1/16 (before 5 pm): HW 1.1, 1.2 (see Current Assignment, Submit problems)
Wed 1/17: Lab 1.1 Memorization (p. 144) and Extra Credit 1.2:22
Mon 1/22: HW 1.3, 1.4
W 1/24: Lab 1.3 Harvesting (p. 146)
Wednesday, January 10, 2007 at 17:58
Warning: Lab 1.1 (p. 144), which is due next class (after MLK day), requires some preliminary work and data collecting from each lab partner before the actual work on the project can start. Try to do as much lab work as you can before the holiday, so that you do not panic Tue evening.
Read Sections 1.3 and 1.4.
1.2: 1-3, 6-10, 15, 19-21, 22x, 23, 25, 27, 33, 35, 38. Submit: 2, 10, 20, 33, 35.
Wednesday, January 17, 2007 at 16:53
Complete the handout Slope Fields and Euler's Method.
Re-read Sections 1.3 and 1.4.
Play with the EulersMethod tool in DETools. Choose an equation, display the slope field, select Compare All, and use the mouse to pick an initial condition. Observe how:
- Euler's method really yields a "broken" line, which crosses the slopes;
- lowering the step size yields better approximations (still broken lines though);
- the "exact" solution is given, in fact, by another (better) numerical method (RK4).
1.3: 1-3, 6, odd 7-13, 14-18, 24x. Submit: 14, 18.
1.4: 1-6, 12, 13, 15, 20. Submit: 5,6,12,13, 20.
Read 1.5, 1.6.
Start work on Lab 1.3 Harvesting (p. 146) with a different partner.
Mon 1/22: HW 1.3, 1.4, and Extra credit 1.3:24.
Monday, January 22, 2007 at 15:36
Use HPGSolver in DETools to solve the IVP y' = -1/y, y(-1) = 2. Explain what happens.
1.5: odd 1-7, 10, odd 13-17, 18. Submit: 10, 18.
Read 1.6 (it is fairly long, but should feel somewhat familiar); we will discuss it next time.
Wednesday, January 24, 2007 at 16:01
Finish the linearization of the model in HW 1.1:16, which we started in class.
1.6: 1, 3, 5, 7-9, 13, 15, 17, 19-24, 29-31, 33, 35, 37, 39, 40, 43x, Animation 1. Submit: 8,20, 22, 40.
Read 1.7.
Start work (with a different partner) on the Bifurcation Lab.
Monday, January 29, 2007 at 18:35
Look up the word bifurcation in a dictionary.
1.7: 1-3, 9, 12-14, 17, 20-23. Submit: 2, 12, 14, 20, 22.
Extra credit: Bifurcation Diagram Shape
1. Prove that the bifurcation diagram for the 1-parameter family y'=y(1-y)-h is indeed a parabola, by deriving an explicit formula for it, in terms of y and h.
2. When does the shape of the bifurcation diagram match the shape of the right-hand side? (Warning: the answer is not "Always"; as evidenced by the family y' = (h-y)y.)
Read 1.8 and 1.9.
Wednesday, January 31, 2007 at 19:12
1.8: 1, 3, 5, 9, 12-17, 21, 25, 33, 34x. Submit: 12, 16, 25.
1.9: 3, 9, 13, 19, 20-15. Submit: 20, 24.
Read 2.1.
W 2/7 (not Monday!): HW 1.7, 1.8, 1.9.
Note: Lab 2.2 Competition is postponed until further notice.
Monday, February 5, 2007 at 19:01
1.8 Extra credit. In order to solve the linear non-homogeneous differential equation y' = -2y + exp(-2t), the textbook advises you to guess y = A t exp(-2t) as a particular solution (to determine the unique value of the constant A, you just plug your guess into the DE). Explain the reason for this "unnatural" second guess, following these steps:
1. Explain why the "natural" first guess y = A exp(-2t) is a hopeless waste of time.
2. Considering a whole family of differential equations y' = -2y + exp(rt), where the DE in question uses r=2, and assuming that r is some exponent different from -2, find the constant A_r (of course, your answer should depend on r) in the "natural" guess y_r = A exp(rt) for each r.
3. What happens to your answer as r gets very close to (but not exactly equal to) -2?
4. Take a limit as r approaches -2 and recognize it as a derivative of some expression at some point (you need to recall the formal definition of the derivative as a limit). Use Calculus to compute this derivative and draw final conclusions. Note that you did not really prove anything (taking the limit of solutions to a differential equation family is not justified by any theorem we mentioned in this course!); now you just have a better reason to expect a solution in the "unnatural" form of y = A t exp(-2t).
Read 2.1 on your own (we'll discuss it next time).
2.1: 1-7, 9-11, 15, 17, 19-21, 23-29. Submit: 1, 4, 6, 17, 25, 26, 28.
Note: complete 2.1: 19, 20 before next class.
Extra credit: Prove that attaching two springs with the same stiffness to a mass, "in parallel," is equivalent to a single spring of double stiffness. Prove also that connecting the same two springs "in series" (that is, making a double-length spring) is equivalent to halving the stiffness.
Read 2.2.
Start work on Lab 2.1 Spring and Magnets.
Monday, February 12, 2007 at 16:35
Read 2.2, 2.3, and 2.4.
2.2: 1-4 by hand, 8, 9, 11-16, 19-23, 26-29. Submit: 8, 11 (explain your decisions), 22.
2.3: 1, 3.
Prove that the solution curves to the system x' = -x, y' = -2y satisfy y = Kx^2 for some K (which depends on the initial condition), hence they are truly (pieces of) parabolas.
Prove that the solution curves to the system x' = x, y' = -y satisfy y = K/x for some K (which depends on the initial condition), hence they are truly (pieces of) hyperbolas.
Wednesday, February 14, 2007 at 16:26
Midterm Test 1 (on Chapters 1 and 2): Wednesday, Feb 21. Look at several sample tests.
2.3: 5-9, 12-14, 15, 17, 19. Submit: 7 with 12, 13 with 14.
2.4: 1, 2, 4, 7-9, 11-15, 16x. Submit: 6, 8.
Read 2.5 on your own (we'll discuss it briefly on Monday) and play with the following tools for the Lorenz system:
- DETools: Lorenz Equations and Butterfly Effect
- IDE tools
- Lorenz Applet by P. Worfolk at UIUC Geometry Center
2.5: 1, 2, 5. Submit: 1, 2, 5.
Start work on Lab 2.2 Species. Use the birthday month of each partner to determine the choice of population systems (p. 226) in the following way: add the month ordinals (1 for January, 2 for February, and so on) and take the remainder when you divide by 4. If the remainder is 0, use choice 4. For example, if Alice was born in June and Bob was born in November, they would add 6 + 11 = 17, divide by 4 and get remainder 1; so they would use the systems in Pair(1).
Monday, February 26, 2007 at 18:00
Review Section 2.3.
Re-read 3.1
Find solutions to the partially-coupled system discussed in class (Problem 3 on the test), which had a matrix (-1 0; 1 2), for the following initial conditions (we used the second one in class; the first one is from the test): (3,0), (0,1), (0,-1), (3,-1), (9,-8). Bring your work to next class.
Play with MatrixField in DETools and the following IDE tools for matrices and vectors before next class. Try to find vectors v such that the input vector v and the output vector Av are colinear (i.e., v and Av point in the same direction or in opposite directions). Try this for different matrices, including all the matrices we used in class: (1 0; 0 -1), (1 2; 3 4), (1 2; 2 4), and (-1 0; 1 2).
1. First, do it using the MatrixField in DETools. Record your observation for each matrix.
2. Next, do it using the IDE Matrix Machine.
3. Finally, do the same using the more precise IDE Eigen Engine (I will explain the strange name in class).
We will discuss your observations next class.
3.1: 5, 9, 14-18, 20-23, 26-29, 33, 34. Submit: 16, 18, 34.
Extra credit: Revisit Problem 14 in 1.7 (Bifurcations).
1. Prove that no such continuous function exists.
2. Describe a discontinuous function that does the job. Make sure that your function works for all parameter values! (Hint: asymptotes.)
3. The requirement for one sink and a source-sink-source triplet is essential. Modify the problem in the following way: remove the equilibria type requirements; in other words, look for a DE family with 1 sink for all a < -1 and 3 equilibria (of any type) for all a > 1. Now find a continuous right-hand side that solves the modified problem.
Wednesday, February 28, 2007 at 18:45
Complete the Test 1 Makeup handout, according to the Honor Code, and submit before 5 PM on on Friday, Mar 2, for grading.
3.2: 1-4, 12, 15-18, 21, 23, 25. Submit: 2, 16.
Does the Linearity Principle apply to the solutions of the second-order differential equation y''+7y'+10y = 0. Why?
Use the MatrixField tool in DETools to find eigenvectors "by mouse" for the matrices in the assignment from last time.
Read 3.3 on your own. We'll discuss it briefly after the break.
M 3/12: HW 3.1, 3.2.
W 3/14: Lab 3.3: Mass in Space.
Monday, March 12, 2007 at 19:18
The Makeup score was multiplied by 0.4 and added to Midterm score.
Midterm Test 1 (with make-up added) statistics: average = 80, median = 79, max = 111, min = 56.
Play with DETool Linear Phase Portraits.
3.3: 1-3, 10, odd 13-19, 20-22, 24, 27. Submit: 2, 20, 24.
Extra credit: 3.3:22.
Read Appendix C on complex numbers and then read 3.4.
Wednesday, March 14, 2007 at 16:01
Read 3.4 again.
3.4: 1, 4, 10, 15-17, 21-25. Submit: 4 with 10, 16, 25
Review 1.8 (both the text and the homework).
Read 3.5 and 3.7.
Wednesday, March 14, 2007 at 18:40
Read 3.4 again.
3.4: 1, 4, 10, 15-17, 21-25. Submit: 4 with 10, 16, 25.
Review 1.8 (both the text and the homework).
Read 3.5 and 3.7.
Watch the animated parameter path in DETools/TDAnimation and in IDE/Path before next class.
Start work on Lab 3.1. Hint: determine five regions in the (a,b)-plane that correspond to the regions in the standard trace-determinant parameter plane.
W 3/21: Lab 3.1: System Bifurcations.
Monday, March 19, 2007 at 18:53
Read 3.5 and 3.7 again.
Watch the animated parameter path in DETools/TDAnimation and in IDE/Path.
3.5: 2, 3, 6, 7, 9-13, 18, 21-24. Submit: 2 with 6, 18, 22.
3.7: 1-3, 8, 11-14. Submit: 2, 14 (all cases: a-d).
Submit: Write a linear system Y'=AY, whose phase portrait consists of the curves x = 0 and y = Kx^3 for all K. Now write a different system that has the same portrait (the same trajectories). What is the same and what is different about the two systems (in terms of eigenvalues, eigenvectors, trajectories, and time plots)?
Complete Lab 3.1. Hint: determine five regions in the (a,b)-plane that correspond to the regions in the standard trace-determinant parameter plane.
Does the Linearity Principle apply to the solutions of the second-order differential equation y''+7y'+10y=0. Why?
Play with the following IDE tools before next class: Mass And Spring, Damped Eigenvalues, Critical Damping Zoom, and Forced Vibrations.
Read 3.6; we'll discuss it in class.
Wednesday, March 21, 2007 at 15:41
Play again with the following IDE tools: Mass And Spring, Damped Eigenvalues, Critical Damping Zoom, and Forced Vibrations.
3.6: 1-3, 7, 15-17, 23-25, 29, 30, 32, 36, 38-40. Submit: 16 with 24, 32, 36, 38, 40.
Extra credit: 3.6: 34, 35.
Extra credit (1.8 revisited). In order to solve the second-order differential equation x'' + 2x' + x = 0 (a critically damped oscillator with a double eigenvalue of -1), the textbook advises you to make a second guess of the form x = C₂ t exp(-t), appealing to Section 1.8. Explain the reason for this "unnatural" second guess, following these steps:
1. Explain why the "natural" first guess x = C₁ exp(-t) is not enough to generate all solutions.
2. Now consider a one-parameter family of differential equations, where each DE has eigenvalues -1 and -1+h and h is a (small) parameter. Write a differential equation with these two eigenvalues.
3. Verify that the choice h = 0 yields the original equation.
4. For each non-zero value of h, solve the IVP with initial conditions x(0) = 0, x'(0) = 1. Make sure your answer satisfies both the DE and the initial conditions.
5. What happens to your answer above as h approaches 0? Take a limit and recognize it as a derivative of some expression at some point (you need to recall the formal definition of the derivative as a limit). Use Calculus to compute this derivative and draw final conclusions. Note that you did not really prove anything -- taking the limit of solutions to a differential equation family is not justified by any theorem we mentioned in this course! Nevertheless, now you have a better reason to expect a second solution in the "unnatural" form of x = C₂ t exp(-t).
6. Finally, prove that if x(t) = p(t) exp(rt), where p(t) is some polynomial of degree n, then x'(t) = q(x) exp(rt), where q(x) is also a polynomial of degree at most n. Now prove that the same holds for the LHS of any second-order linear homogeneous DE, that is for a x'' + b x' + c x. You have now justified the choice x(t) = p(t) exp(rt) as a (second) guess in any linear homogeneous DE.
Review 1.8.
Read 4.1 and 4.2; we'll discuss them next time.
Monday, March 26, 2007 at 18:28
In my excitement over linearity principles and periodic forcing, I messed up the second way to find a particular solution in class: namely, I forgot to "complexify" the external force (I should have replaced sin(t) in the RHS with e^it), so my botched computation (albeit correct) was useless and not impressive at all. Please follow the correct procedure described on p. 398. Mea culpa.
Play with the Forced Oscillator. Set the mass to 0.5, the stiffness to 2, and the damping to 0 (undamped). Estimate the natural frequency/period of the oscillator from the graphs. Confirm your estimate by computing the exact frequency (using the numerical values of m, and k). Now set the external frequency w to 1.8 (which is near the natural frequency). Observe the beating of the solution (watch only the yellow position).
4.1: 1, 2, 9, 13, 14, 19-21, 26, 31, 32, 41. Submit: 14, 20, 26, 32.
4.2: 3, 4, 11, 16-20. Submit: 4, 20.
Wednesday, March 28, 2007 at 16:43
Start work on Lab 5.1 Magnets with a Kick (see due date below).
4.3: 1, 4, 5, 10, 15, 19, 21. Submit: 10, 15.
Observe beating in the Forced Oscillator. Set the mass to 1/2, the stiffness to 2, and the damping to 0 (undamped). Estimate the natural frequency/period of the oscillator from the graphs. Confirm your estimate by computing the exact frequency (using the numerical values of m, and k). Now set the external frequency w to 1.8 (which is near the natural frequency 2). Observe the beating of the solution (watch only the yellow position).
Study the resonance and near-resonance (beating) behavior of the Forced Oscillator for different initial conditions. What happens if you add small damping?
Extra credit. In order to find a particular solution to x'' + x = cos t (an undamped oscillator forced at resonance frequency), the textbook advises you to make a second guess for of the form x = a t exp(it) on p. 416, appealing to Sections 1.8 and 4.1. Explain the reason for this second guess, following these steps:
1. Explain why the "natural" first guess x = a exp(it) or x = A cos(t) + B sin(t) is hopeless.
2. Now consider the one-parameter family of forced oscillators x'' + x = cos((1+h)t), where h is a small parameter. What is the physical meaning of h?
3. For each non-zero value of h, find a particular solution of the form A cos((1+h)t).
4. Using your particular solution above, find the unique solution to the IVP x'' + x = cos((1+h)t) with initial conditions x(0) = x'(0) = 0. Make sure your answer satisfies both the DE and the initial conditions.
5. What happens to your answer above as h approaches 0? (Hint: expand the denominators in your solution and factor out h.) Take a limit and recognize it as a derivative of some expression at some point (you need to recall the formal definition of the derivative as a limit). Use Calculus to compute this derivative and draw final conclusions. Note that you did not really prove anything -- taking the limit of solutions to a differential equation family is not justified by any theorem we mentioned in this course! Nevertheless, now you have a better reason to expect a particular solution in the "unnatural" form x = A t sin(t).
6. Finally, prove that if x(t) = p(t) exp(iwt), where p(t) is some polynomial of degree n, then x'(t) = q(x) exp(iwt), where q(x) is also a polynomial of degree at most n. Next, prove the same for x''(t). Now prove that the same holds for any periodically-forced oscillator, that is for m x'' + b x' + k x = exp(iwt). You have now justified the choice x(t) = p(t) exp(iwt) as a (second) guess in any second-order linear DE.
Review 2.1, 2.2 and read 5.1.
Read Chapter 3 of the award-winning 2004 Honors Thesis of Shannon Jessie (PDF: 2 MB).
Monday, April 2, 2007 at 17:08
Play with the IDE tools Pendulum (try Linear, Simple and Damped Nonlinear), Van der Pol Nonlinear Oscillator, and the Competitive Species Model.
5.1: 1-3, 5-7, 17-20, 27-29. Submit: 2, 6, 20, 28.
Extra credit: Use the Chemical Oscillator DETool and assume that a=10 in all parts.
1. Verify that (2,5) is the equilibrium (hint: plug it in).
2. Linearize the system (compute its Jacobian).
3. Determine the equilibrium type for b=1 and b=6. Verify your answer in the DETool.
4. Approximate the bifurcation value b* by trial-and-error in DETools.
5. Find the bifurcation value b* exactly (hint: look at the trace of the Jacobian).
6. Show that when b has the bifurcation value, the linearized system has a center.
7. Despite that, show that, for the same value of b, the original system has a spiral (albeit somewhat sluggish) sink. This is called a Hopf Bifurcation.
8. Read more about this reaction in Problem 2 (PDF) from a UCSF course in Biophysics.
Read Chapter 4 of the award-winning 2004 Honors Thesis of Shannon Jessie (PDF: 2 MB).
Wednesday, April 4, 2007 at 20:25
Start work on Lab 5.3.
5.2: 3-6, 15, 17, 21-23. Submit: 4, 6.
5.3: 1, 4-8, 10, 12, 15*, 16x, 18x. Submit: 4, 6, 8, 10.
Extra credit: 5.3:16, 18.
I wrote up a short Maple worksheet to plot the Hamiltonian for the undamped pendulum. You can interact with this worksheet in any campus computer lab (click inside the 3-D plot with the mouse and drag it around to tilt and rotate the view). From home, you can still see the Hamiltonian (but can't change the 3-D view) in a 3-D plot, in a 2-D contour plot, or see the contours floating in 3-D. Enjoy.
Observe how the total energy is conserved along the trajectories of the undamped oscillator.
Now you have all the tools to completely understand (and appreciate) the award-winning 2004 Honors Thesis of Shannon Jessie (PDF: 2 MB). Read again Chapters 2-4, armed with your new knowledge of nullclines, Jacobians, and Hamiltonians. Pay special attention to Lemma 4.2 (p.38), as well as its clever use at the end of Sections 6.1.2 (p.48) and 6.2.3 (p.53).
Monday, April 9, 2007 at 19:04
8.1: 1-6, odd 9-15, 22, 23, odd 27-31, 33-39. Submit: 12, 14, 36.
Read Chapter 6 of the TI-83 Guidebook (PDF 4MB) or browse the list of guidebooks for TI graphing calculators. Learn how to plot time series of orbits on your calculator before next class.
Thursday, April 12, 2007 at 14:11
As you know, the Honors College Symposium will be held this Friday, 4/13. Attend the student talks and view the posters (check the schedule). This would be a great opportunity to see what you senior fellows have done; you will be doing the same in your senior year.
Play with the BU Web diagram applet. You can use it in the exercises below.
8.2: 1-3, 9, 13-15, 18-20, 29, 30. Submit: 2, 14, 20, 30.
Review Section 1.7 Bifurcations.
Read ahead 8.3
Monday, April 16, 2007 at 19:28
Because of an earlier typo, I postponed the Lab 5.2 due date (see below).
Reminder: Midterm Test 2 (on Chapters 3-5) is take-home. You can pick up a copy from my mail box on Friday, 4/20. The complete test is due Monday, 4/23, in class.
Look at several sample tests.
Play with the BU Web diagram applet. You can use it in the exercises below.
Review Section 1.7 Bifurcations.
8.3: 2-4, 8, 10.
Extra credit: 8.3: 10. Hint: Compute explicitly the 2-periodic points of F, by dividing out the factors of F²(x) - x that come from the fixed points of F (which are, of course, fixed points of F² as well).
Read ahead 8.4.
Start work on Lab 5.2 Pendulum. For simplicity, assume the the pendulum length is 32 ft, so that the nonlinear equation is simply x'' = -sin x and the linear pendulum x'' = -x has period close to 6.3 seconds. (Such a large pendulum is often used to demonstrate in a non-trivial way the Earth rotation and is called Foucault's Pendulum.) Unfortunately, the HPGsystem tool in DETools does not have tick marks in the time plots, so estimating oscillation periods is difficult. Here are two options:
1. In DETools, use a large time interval (say t = 62.8 ~ 20 pi) and count carefully the number of periods within it, with precision 1/4 of a period. For example, I counted 7.5 periods from t=0 to t=62.8 when x₀=2 (of course, the linear pendulum will have exactly 10 periods from 0 to 20 pi -- why?).
2. Instead of DETools, use the PPLANE applet by Polking (wait for it to download fully, click inside the frame with the two buttons, and then press the right button labeled PPLANE; several windows will pop up). You can click an initial condition with the mouse directly in the Phase Plane window or key an exact number from the keyboard, after pressing Ctrl-K. To get time plots for a computed trajectory, follow menu Graph, x vs. t. You can't measure the periods directly either, but you have tick marks here. As suggested above, measure several consecutive periods to get better precision.
In either case, provide explicitly the fraction you use to compute the periods (in my example, the period would be 62.8/7.5 = 8.4).
The last lab assignment is an evaluation essay, which is due Wed, 4/25.
W 4/18: HW 8.1-8.2
M 4/23: Lab 5.2 Pendulum.
W 4/25: Last Lab: Reflections on the Course.
Wednesday, April 18, 2007 at 18:17
Midterm Test 2 (take-home) statistics: average = 92, median = 96, max = 106, min = 60. Wow!
The discussion topic on Mon, 4/23, is up to you. Here are the choices:
1. Complete our study of chaos and tie it with the Lorenz attractor. Sections: 2.5, 5.5, 8.5.
2. More nonlinear systems: 5.4
3. Systems in 3-D: 3.8 and 5.5
4. How does DETools compute orbits? Numerical methods beyond Euler: 7.2-7.3
5. Other topics we have not discussed (outside of Chapter 6).
Please email me your opinion by Sunday noon, so that I can prepare.
Final review session on Wed, 4/25, 2-3:30pm. Send me questions and problems to review.
Final test: Fri, 4/27, 1:15-3:45 pm.
8.4: 2-5, 12-14.
8.R: 1-16, 20.
Friday, May 4, 2007 at 13:31
Final test statistics: average = 84, median = 83, max = 98, min = 60. Well done!
If you are curious, you can stop by my office and peek in your final test (we are required to keep all finals on record).
Have a great summer, full of exciting non-linear dynamics!
The book Nonlinear Dynamics and Chaos, by Steven Strogatz, ISBN-13: 978-0738204536 is an excellent post-course reading. It goes deeper into a lot of the topics we discussed and illuminates many issues that we only touched upon. Even though it is more advanced than our textbook, you are well prepared to read it.

Last Updated: Friday, May 4, 2007 at 13:31

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Eugene Belogay