Ordinary Differential Equations
MATH 304, Exam #1
Wednesday, Oct. 4, 6:00 - 7:30 pm, Swords 328

The first exam covers all of Chapter 1 of the course text. You should go over homework problems and your class notes. I have also chosen some good practice problems from the Chapter 1 review. The answers to the odd numbered questions are in the back of the book while the answers to the evens are given below. The exam will be designed to take one hour although you will have the full 90 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 a computer.

Exam Review: We will review for the exam during Monday's class, Oct. 2nd. Please come prepared with specific questions.

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

  1. General ODE terminology: general solution, particular solution, initial-value problem, initial condition
  2. Types of ODE's: order of, autonomous, linear, separable, homogeneous, nonhomogeneous, periodic
  3. 1st-order ODE's - analytic techniques: separation of variables, guess and test (method of undetermined coefficients), integrating factors, Linearity principles, mixing problems
  4. 1st-order ODE's - numerical techniques: Euler's method, slope fields
  5. 1st-order ODE's - qualitative ideas: equilibrium solutions (source, sink, node), phase lines, long-term behavior, bifurcations, bifurcation diagrams, Existence and Uniqueness Theorems
  6. Periodic Differential Equations: periodic solutions, the Poincare map, fixed points
  7. Population models: unlimited growth model, logistic population model (with constant or periodic harvesting), carrying capacity, predator-prey systems

Chapter 1 Review Exercises (pp. 138 - 143)
Problems:   1, 2, 4, 7, 9, 11, 17, 19, 21, 23, 26, 27, 29, 35, 37, 39, 41, 43, 47, 48, 49, 51, 53, 54, 55, 57

The answers to the evens are:
2.   y(t) = y_0 e^(3t)
4.   y = k pi, for any integer k.
26.   (a) linear, homogeneous, (b) y(t) = 1 + t + (1+t)^2 ln|1+t| + k(1+t)^2
48.   y_4 = y(2) = 1.810 which does not make sense because the solution should stay below y = sqrt(3), which is an equilibrium solution.
54.   (a) Exponential model: P(t) = 5.2 e^(0.383 t)   Logistic model: k = 0.383, N = 210 (turns out to be a good fit.) (b) While the exponential model is not a good fit, the logistic model is. (c) p = 186.5 in our logistic model. (d) Cell phone subscriptions appear to be leveling out around 210. The cell phone industry will become more competitive and cell phone prices should decline.

Additional Problem: Sketch the bifurcation diagram for the family of differential equations dy/dt = y^3 - ay with a as a parameter and describe the behavior of solutions before, at and after any bifurcations. Explain why this bifurcation is called a pitchfork bifurcation.