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
13.
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
