# MATH 250b

## Spring Semester 2008, Section 2, Professor J. M. Cushing

SYLLABUS Math 250b ( Spring Semester 2008 )
Chapters 1-6 from Differential Equations: an Applied Approach by J. M. Cushing
Pearson Prentice-Hall,
Upper Saddle River, New Jersey, 2004 (ISBN: 0-13-044930-X)

Subject to change

PRELIMINARIES & MODELING METHODOLOGY
Introduction 1.1 - 1.2
Model derivation (variable identification/classification/symbolization/inter-relationships, laws/assumptions/hypotheses)
Mathematical analysis (solution formulas, analytic/numeric/graphic approximations, qualitative methods)
Solution interpretation/utilization (evaluation/critique/data)
Model modification (re-evaluation of assumptions, added phenomena & mechanisms)
1. Thursday, January 17

2.
Tuesday, January 22

3.
Thursday, January 24

FIRST ORDER EQUATIONS
1. Fundamentals :  Chapter 1
Solutions, the Fundamental Existence & Uniqueness Theorem, graphic approximations, numeric approximations
4. Tuesday, January 29

5.
Thursday, January 31

2. Linear First Order Equations :  Chapter 2
General solution & initial value problems, Variation of Constant formula, homogeneous & non-homogeneous equations, method of undetermined coefficients, autonomous equations (equilibria, stability, phase line portraits)
6. Tuesday, February 5
7.
Thursday, February 7

3. Nonlinear First Order - Autonomous equations (qualitative theory & methods) : Chapter 3.1
Phase line portraits, equilibria, attractors/repellers/sinks, linearization, qualitative equivalence, dependence on parameters & bifurcations, bifurcation diagrams
8. Tuesday, February 12

9.
Thursday, February 14

10. Tuesday, February 19

11.
Thursday, February 21

4. Nonlinear first Order - Non-autonomous equations (analytic methods) :  Chapter 3.2 - 3.4
Solution formulas (separable equations, change of variables), approximation formulas (Taylor polynomial methods)
12. Tuesday, February 26
1
3.
Thursday, February 28

14. Tuesday, March 4

15.
Thursday, March 6 (TEST #1)

SYSTEMS & HIGHER ORDER EQUATIONS
1. Fundamentals :  Chapter 4.1
Conversion of higher order equations to systems, Fundamental Existence & Uniqueness Theorem,
graphic & numeric approximations, phase plane
16. Tuesday, March 11

2. Linear Systems - Basics :  Chapter 4.2
Structure of general solution, initial value problems
17. Thursday, March 13

SPRING BREAK: March 15-23
3. Linear Systems – Autonomous Homogeneous :  Chapter 5.1 - 5.5
Solution formulas (using eigenvalues), short cuts for 2nd order equation, construction & classification of phase plane portraits (nodes, saddles, spirals, centers), stability
18. Tuesday, March 25
19.
Thursday, March 27

20. Tuesday, April 1
21.
Thursday, April 3

22. Tuesday, April 8
23.
Thursday, April 10

4. Linear Systems – Autonomous Non-homogeneous :  Chapter 6.1 - 6.2
Variation of Constants formula, method of undetermined coefficients for 2nd order equations
24. Tuesday, April 15

5. Autonomous Nonlinear Systems – Equilibria :  Chapter 8.1 - 8.3
Equilibria, stability, linearization, Fundamental Theorem of Stability, geometry of local phase portraits
25. Thursday, April 17
26.
Tuesday, April 22

27. Thursday, April 24  (TEST #2)

6. Autonomous Nonlinear Systems – Oscillations :  Chapter 8.5
Periodic solutions, limit cycles, bifurcations & Hopf criteria
28. Tuesday, April 29

29.
Thursday, May 1

Review
30. Tuesday, May 6

J. M. Cushing  / Department of Mathematics  / Program in Applied Mathematics  / University of Arizona / Tucson, AZ, 85721-0089

(last revised 28 January 2008)