MATH 237- HONORS DIFFERENTIAL EQUATIONS I-COURSE LOG

8/20- Syllabus
Initial-value problems; explicit vs. implicit solutions, domain of definition (Lesson 3)
HW 1 (p.27): 3, 4(a): for the initial-value problem y(-2)=-1, find the domain of the solution
4(b): for which (x_0, y_0) does  a solution with y(x_0)=y_0 exist?
(due Thursday 8/27)

8/25-Lesson 4: n-parameter families of solutions vs. "general solution"
HW 2(p.37): 7, 13, 19, 20 (due 9/3)

8/27: Graphical analysis of autonomous equations (online handout)
Existence/uniqueness theorem
HW 2: problems on the handout (due 9/3)

9/1: Autonomous equations: E/U theorem, finite-time blowup, breakdown of uniqueness

9/3: Solution of HW1/ Lesson 6C: first-order equations in differential form, separable differentials.
HW 3: (p.55): 4, 7, 12, 19, 20 (due 9/10)

9/8: Equations with homogeneous coefficients (Lesson 7), linear coefficients (Lesson 8B)
HW 3: p.61 5,6,15/ p. 69 9,12 (due 9/10)

9/10 Exact equations, integrating factors. (9A, 9B, 10B)- taught by Brian Allen
HW 4: p.79 5, 6, 17 (due 9/17)

9/15 1st order linear equations, Bernouilli equations (Lesson 11)-taught by Jochen Denzler
HW 4: p. 97: 3, 6, 21, 23 (due 9/17)

9/17 Riccati equations, Miscellaneous methods (Lesson 12)-taught by Jochen Denzler
HW 5 p. 103 3, 9, 11 (due 9/24)

9/22 Review
HW 5 p. 98: 28, 29 (due 9/24)

9/24 Applications: Lessons 13, 17

9/29 TEST 1.
Test 1 (with solutions)

10/1 Applications: Lesson 17 problems
HW6 p.120 9, 12/ p. 176, 4/p. 192, 34 (due 10/8)

10/6 Linear second-order equations: homogeneous (Lesson 20)

10/8 linear second-order equations: double root, non-homogeneous (undet. coefficients) (Lesson 21A)
(taught by Jochen Denzler)

10/13 class canceled

10/15 FALL BREAK

10/20 linear second-order equations: use of differential operators, variation of parameters (Lesson 22)
HW 7: p. 231: 6,8,21/ p.240: 1,16 (due 10/22)

10/22 linear equations with non-constant coefficients: reduction of order (Lesson 23)
variation of parameters formula/ Equations of Cauchy-Euler type (not in text)

HW8 (due 10/29) p. 246: 3, 5, 19 (solve as Cauchy-Euler--general solution)
Problem 23: solve using the theory in problem 22 (transformation to Riccati equation).
Problem 25: use the theory in this problem to solve the example given (exact second-order equation)
EXTRA CREDIT: prove the assertions made in problems 22 and 25. (5 points per correct solution, added
to grade in exam 2 or exam 3.)

10/27 Laplace Transform
Practice problems: p.311: 2,3,6,7,15,16,17

10/29 Laplace Transform
Topics on second-order equations
(Topics seen in lecture, but not easily found in the text; includes practice problems.)

11/3 TEST 2, material included: lectures from 10/1 to 10/29 (including handout). Emphasis
on homework and practice problems.
Test 2 (with solutions)

11/5 Laplace transforms: periodic functions, Gamma function, convolution theorem
HW 9( due 11/12): p.311--22(c),(d), 24.
For practice problems on periodic functions and the convolution theorem, see the handout posted on 10/29 (above).

11/10 Lesson 34: motion in a conservative vector field, properties of central vector fields, example F=-kr

11/12 Lesson 34: inverse-square central field, planetary motion
HW9 due

11/17 Lessons 34/35: From Kepler's laws to the inverse square law/ Special 2dn order equations
HW10: last two problems on the handout of 20/29, and: p. 480 no.7, p492 no.1, p.497 no 7,8.

11/19 Lesson 35 (nonlinear second-order equations of special type
Practice problems (from text, p.504): 1, 8, 14, 18, 19

11/24 TEST 3: Lectures from 11/5 to 11/19
Test3 (with solutions)

11/26 Thanksgiving Holiday (no classes)

12/1 Lesson 31 (last day of classes): non-degenerate first-order 2X2 systems/ number of parameters in the general solution,
solution by reduction to "triangular form". (Lesson 31D, E in text)
Practice problems: p. 421: 5, 6, 8, 12.

FINAL EXAM: Thursday, Dec. 10, 8AM--10AM (comprehensive)