Math 321
Differential Equations

Instructor

Ernest Boyd, Wissink 273, 389-1452
Office Hours: 8:15 - 8:50 am, MTWHF; 9:00 - 9:50 am, F; 10:00 - 10:50 am, MWHF; 11:00 - 11:50, W.

Textbook

Elementary Differential Equations, 8th Edition, by Boyce and DiPrima

Course Content

First order linear and nonlinear ODE's. Second order linear ODE's. N-th order linear ODE's with constant coefficients. Variation of Parameters, Undetermined Coefficients, Laplace Transforms and Numerical Algorithms. Two-dimensional linear and nonlinear systems with Poincare-Bendixson Theory. Familiarity with a computer algebra system, Mathematica.

Schedule

11:00 - 11:50 am, MTHF, Wissink Academic Computer Center Room 116

No class on Friday, Nov. 9.

Tests

Fri, Sep. 21

Mon, Oct. 22

Tue, Nov. 20

Final Exam

Tuesday, Dec. 11, 10:15 am - 12:15 pm

Grading Policy

Each test will be graded on its own scale normalized to 100%. The final will also be graded on a scale normalized to 100%. All of the homework and quizzes collected will be totaled and graded as one test. At the end of the semester the five grades will be averaged and compared to the average of the scales. Grading will reflect the mathematical correctness of the answer and the quality of the presentation in terms of completeness and organization. Late assignments will be penalized. Assignments over one week late will receive no credit. Active participation in the class discussions will also be factored into the evaluations. Make-up tests will be given only in special cases, when an excused absence has been prearranged. Incompletes will follow the university's policy as expressed in the student bulletins and the faculty handbook. No incomplete will be given to repeat the entire course.

MSU provides students with disabilities reasonable accommodation to participate in educational programs, activities or services. Students with disabilities requiring accommodation to participate in class activities or meet course requirements should first register with the Office of Disability Services, located in 0132 Memorial Library, telephone 389-2825, TDD 711 and then contact me as soon as possible.

Mathematica Supplements

1. Basic Example of Differentiation and Integration
2. Trigonometric Identities
3. Direction Field of y' = f(t, y)
4. Logistic Equation
5. Picard Iteration Method
6. Second Order Homogeneous Equation
7. Wronskian
8. Amplitude Modulation
9. Variation of Parameters Method
10. Phase Plane
11. Competition Model
12. Lotka-Volterra Predator-Prey Model
13. Lotka-Volterra Predator-Prey Model in Dimensionless Form
14. Two dimensional system
15. Van der Pol oscillator
16. Pendulum without driving force
17. Pendulum with driving force
18. Laplace Transform Summary Sheet
19. Laplace Transform Mathematica
20. ODE's with Square Waves

Homework

1. Page 8 #24. Page 17 #12-14. Hand in these four problems on Friday, Sep. 7.
2. Page 39 #1, 8, 11, 18, 24, 38. Hand in #26 on Tuesday, Sep. 11.
3. Page 47 #3, 7, 9, 16, 24, 27. Hand in #24, 27 on Friday, Sep. 14.
4. Page 63 #18, 19. Hand in #18 on Monday, Sep. 17.
5. Review Problems
6. Page 142 #23, 25. Hand in #23, 25 on Tuesday, Oct. 2.
7. Page 164 #24. Hand in #24 on Tuesday, Oct. 2.
8. Page 172 #17. Hand in #17 on Tuesday, Oct. 2.
9. Page 151 #9, 14, 26. Page 158 #24, 25.
10. Page 184 #2, 5, 18, 24.
11. Click here for the assignment. Hand in these problems on Tuesday, Oct. 9.
12. Page 203 #9. Page 214 #6, 8, 17. Hand in these problems on Friday, Oct. 12.
13. Page 190 #5, 7, 17, 18.
14. Review Problems
15. Page 312 #19, 26, 27.
16. Page 322 #1-10, 21-23, 33, 37. Hand in #8, 10, 23 on Friday, Nov. 2.
17. Page 329 #4, 6, 9, 11, 15, 17, 28, 30, 31. Hand in #9, 17 on Thursday, Nov. 8.
18. Page 337 #2, 3, 10, 16. Hand in #16 on Tuesday, Nov. 13.
19. Page 344 #4, 15, 16. Hand in #15 on Thursday, Nov. 15.
20. Use Laplace transforms to solve y'' - 2y' + y = t³e^t with y(0) = -1 and y'(0) = 2.
21. Use Laplace transforms to solve y'' - y = e^tcos(2t) with y(0) = 0 and y'(0) = 0.
22. Express L(t*y'(t)) and L(t^2*y''(t)) in terms of L(y(t)).
23. Prove for sufficiently large p that Gamma'(p) < Gamma(p+1) and then Gamma(p) < C*exp(p^2/2) for some constant C.
24. Calculate the Inverse Laplace Transform for 3s/(s^2 - s - 6), (2s-3)/(s^2-4), (8s^2-4s+12)/s/(s^2+4), (1-2s)/(s^2+4s+5)
25. Solve y'' + y' + y = sin t + u(t - π/2)*e^{-(t - π/2)}, y(0) = 1, y'(0) = 0.
26. Find the constant A so that the solution of y" + 4y = Aδ(t - 5π/2) with y(0) = 1 and y'(0) = 0 satisfies y(23π/8) = 1.
27. Review
28. Page 398 #2, 4, 6, 24-29. On Thursday, Dec. 6, hand in the following problem: Solve x' = -4x - 3y and y' = 2x + 3y. Draw the phase plane with two trajectories similar to pages 392-393. Pick your initial conditions so that one trajectory is on one side of the separatrix and the other trajectory is on the other side of the separatrix.
29. Page 410 #2, 3, 4, 14. On Thursday, Dec. 6, hand in #14 including a trajectory in each phase plane to illustrate the general behavior.
30. For matrix A = {{-3,1},{3,-1}} find the general solution of v' = Av and determine the limit of v(t) as t -> Infinity as it depends on the initial conditions.
31. Set up Q'' + 2Sqrt(2)Q' + 2Q = 0 as a two-dimensional system and determine the general solution. What is the direction of asymptotic approach to the equilibrium?
32. Page 428 #4. Try to pick initial values spaced around the perimeter of the region to show the behavior from different sections of the phase plane.
33. Page 493 #17, 20, 21. You need to recognize that the characteristic equation we used in Chapter 3 to determine the characteristic values is the same as the equation for the eigenvalues of the two-dimensional system.
34. Page 534 #1.
35. Page 525 #2.
36. Page 502 #7 and page 511 #8 are the same problem.
37. Review