# Math 251H - Spring 2011

Schedule of classes :
MTW F 04:40P - 05:30P in 115 OSMOND (Schedule n. 559501)

Office Hours (subject to change): TUESDAY 5:45- 6:45 PM, THURSDAY 5 - 6 PM, FRIDAY 1:15 - 2:15 PM.
Finals week: MONDAY MAY 2 and TUESDAY MAY 3, 11 AM - 12 PM

Final Exam: TUESDAY, MAY 3, 2:30P-4:20P 114 AG ENGR (Comprehensive, List of topics)
Review Problems
Practice exam for Final
Practice exam for Final, skip #4 (from Professor Xianto Li's Math251H Course)
Math 251 Sample Exams with solutions.

MIDTERM 2 will cover Sections 3.3 - 7.5 included (List of topics), but not 7.4.
Review Problems
Sample Midterm 2 exam (from Professor Xianto Li's course), with solutions.

MIDTERM 1 will cover Sections 1.1 - 3.2 included (List of topics).
Sample Midterm 1 exam (from Professor Xianto Li's course). Skip problems 5, 6.

Chapter Summary Sheets (from textbook companion website):

Chapter 1.
Chapter 2.
Chapter 3.
Chapter 6.
Chapter 7.
Chapter 9.
Chapter 10.

FINAL EXAM REVIEW: Friday April 29 in class.
QUIZ 9: TUESDAY April 26 (Sections 9.2, 9.3, 10.1).
QUIZ 8: MONDAY April 18 (Sections 7.6, 7.8, 9.1).
MIDTERM 2 REVIEW: Tuesday April 5 in class.
QUIZ 7: FRIDAY April 1 (Sections 7.1, 7.2, 7.3, 7.5).
QUIZ 6: FRIDAY MARCH 25 (Sections 6.3-6.6).
QUIZ 5: WEDNESDAY MARCH 16 (Sections 3.7-3.8-6.1-6.2).
QUIZ 4: FRIDAY FEBRUARY 25 (Sections 3.3--3.6 included).
QUIZ 3: TUESDAY FEBRUARY 15 (Sections 2.6, 2.8, 3.1, 3.2).
QUIZ 2: FRIDAY FEBRUARY 4 (Sections 2.2-2.5 included).
QUIZ 1: FRIDAY JANUARY 21 (Sections 1.1-2.1 included).

Syllabus.

Course Topics: from Elementary Differential Equations and Boundary Value Problems, by Boyce and Di Prima, Wiley, 9th Ed.
1. Introduction: mathemtical models and direction fields (Section 1.1), some simple ODEs and their solutions (Section 1.2), ODE classification (Section 1.3).
2. First-order ODEs: integrating factors (Section 2.1), separable equations (2.2), examples (2.3), Linear and nonlinear eqs (2.4), autonomous eqs and population dynamics (2.5), exact equations and integrating factors (2.6), existence and uniqueness theorem (2.8). For now, skip Euler's method (2.7) and Difference eqs (2.9). Also, skip Logistic growth with a threshold (in 2.5.).
3. Second-order ODEs: Constant-coefficient, homogeneous equations (3.1), linear homogeneous equations, the Wronskian (3.2), Complex roots (3.3), repeated roots and reduction of order (3.4), Non-homogeneous equations and the method of undetermined coefficients (3.5), Variation of Parameters (3.6). Cover some parts of Section 3.7 (mechanical and elettrical oscillations) and 3.8 (forced vibrations).
4. Laplace Transform: Definition (Section 6.1), Solution of the Initial Value Problem for ODEs (6.2), Step functions (6.3), ODEs with discontinuous forcing (6.4), Impulse functions (6.5), convolution integrals (6.6). We will skip Chapters 4 and 5.
5. Systems of first-order, linear equations: Introduction and examples (7.1), Review of Matrices (7.2), Linear systems of algebraic equations: eigenvalues and eigenvectors (7.3), Systems of linear, first-order ODEs (7.4), Homogeneous, constant-coefficient systems (7.5--7.8), Classification of critical points and the phase plane (9.1), Non-homogeneous systems (7.9). Non-homogeneous systems (7.9). Section 7.9 is a reading assignment, you will not be tested on it. We will skip multiple spring-mass systems in Section 7.6.
6. Nonlinear ODEs and Stability: Summary of phase plane analysis for linear systems (9.1), Non-autonomous systems and stability (9.2), Locally linear systems (9.3), Lorenz attractor and chaos (9.8). We will cover 9.8 *at the end* if there is time.
7. Partial differential equations: boundary-value problems (10.1), Fourier Series (10.2), Heat conduction (10.5), Wave equation and vibrating string (10.6). We will cover only the major concepts in each section.

Homework Problems: solutions will be available on ANGEL ( under the "lessons" tab) roughly one week after the problems are posted.
1. Section 1.2: 1, 3, 6, 8, 12, 13, 17. Section 1.3: 3, 5, 13, 30.
2. Section 2.1: 4, 10, 14, 19, 24, 27, 31, 40. Section 2.2: 2, 5, 11, 14, 21, 24, 36.
3. Section 2.3: 7, 12, 17, 27. Section 2.4: 3, 5, 10, 12, 15, 22, 26, 31, 33. Section 2.5: 2, 5, 12, 15, 27.
4. Section 2.6: 1, 4, 8, 13, 16, 18, 20, 24, 28. Section 2.8: 2, 9, 15. Section 3.1: 4, 6, 13, 18, 22, 27.
5. Section 3.2: 4, 6, 9, 15, 32, 38, 44, 49, 51. Section 3.3: 6, 12, 19, 25, 28, 34.
6. Section 3.4: 7, 12, 18, 19, 29, 32, 38, 40. Section 3.5 : 13, 15, 16, 17.
7. Section 3.6: 4, 7, 17, 20, 21. Section 3.7: 3, 5, 10, 14, 15, 18, 32 a) and b) only. Section 3.8: 2, 6, 13, 16.
8. Section 6.1: 4, 10, 14, 18, 22, 26. Section 6.2: 5, 12, 20, 25. Section 6.3: 11, 15, 23, 25, 37.
9. Section 6.4: 4, 15, 17, 18. Section 6.5: 2, 10, 16, 18, 25. Section 6.6: 2, 6, 11, 17, 22, 28, 29.
10. Section 7,1: 3, 5, 15, 19. Section 7.2: 2, 9, 11, 17, 20, 23. Section 7.3: 4, 5, 13, 19, 24, 30--34.
11. Section 7.4: 3, 4, 7, 9. Section 7.5: 2, 3, 8, 12, 17, 22, 24, 29, 31.
12. Section 7.6: 2, 4, 8, 10, 12 14, 22, 28. Section 7.7: 5, 8, 12, 15, 16, 17.
13. Section 7.8: 4, 6, 8, 18 (a, b, c, d only), 19. Section 9.1: 5, 8, 15, 19, 20. Section 9.2: 4, 6, 11, 19, 24.
14. Section 9.3: 2, 7, 12, 21, 22, 27, 28. Section 10.1: 5, 11, 15, 19, 23.
15. Section 10.2: 3, 4, 8, 16, 18, 27, 28. Section 10.5: 6, 7, 11. Section 10.7: 2, 5, 9, 23.

Direction field and Phase Portrait Applet, by Professor John Polking.
Level Curve Plots (Mathematica© notebook).
Phase Portraits (Mathematica© 8 notebook).
Phase portrait for the damped non-linear pendulum (Mathematica© 8 notebook).