Ordinary Differential Equations

MATH 304-01, Exam #1

Wednesday, Oct. 6, 1:00 - 1:50 pm

The first exam covers the supplementary material from the Blanchard, Devaney, Hall text (Sections 1.2 and 1.8) as well as Sections 1.1 - 1.4 and 2.1 - 2.7 of the course text. You should go over homework problems and your class notes. The exam will be designed to take 45-50 minutes.

Note: You will be given a scientific calculator for the exam which does NOT have graphing capabilities so be prepared to answer questions without your personal calculator or computer. The only numerical computations asked will be doable on a basic scientific calculator.

Exam Review: We will review for the exam on Monday, Oct. 4th from 7:00 - 8:15 pm in Swords 359. Please come prepared with specific questions.

The following concepts, definitions and models are important material for the exam:

  1. General ODE terminology: general solution, particular solution, initial-value problem, initial condition
  2. 1st-order ODE's: separation of variables and integrating factors (our two analytic techniques), mixing problems
  3. 1st-order ODE's: equilibrium solutions, phase lines, slope fields, bifurcations, bifurcation diagrams (the qualitative techniques)
  4. 1st-order ODE's: periodic differential equations, periodic solutions, the Poincare map, fixed points
  5. Population models: unlimited growth model, logistic population model (with harvesting, periodic harvesting, etc.), carrying capacity
  6. Planar Linear Systems: converting 2nd-order equations into planar systems, vector field, direction field, equilibrium points, phase plane
  7. Linear Algebra: linear independence, basis, determinant, eigenvalues and eigenvectors, characteristic polynomials
  8. Planar Linear Systems: straight-line solutions, general solution, linearity principle

Some Practice Problems:
(The answers to the odds for Blanchard, Devaney and Hall are in the back of the book on reserve.)

Blanchard, Devaney, Hall handouts
Section 1.2 :   2, 5, 7, 9, 13, 17, 25, 31, 33, 41
Section 1.8 :   1, 7, 11, 13, 21, 23, 26

Chapter 1 Exercises (pp. 16 - 19)
Problems:   2a, 2d, 3a, 3b, 5, 8, 9

Chapter 2 Exercises (pp. 36 - 38)
Problems:   1b, 6, 8, 10

Additional problem: Consider the population model P' = 2P - P^2/50 (t in years) for a species of fish in Crystal Lake (without fishing). Suppose fishing by humans will be allowed at the lake and each license is allowed to catch 3 fish per year (these are hard fish to find!) How many licenses can be issued if the fish are to have a chance of surviving? What will happen to the fish population if the maximum number of lisences is issued? How does the behavior depend on the initial population?