Ordinary Differential Equations

MATH 304, Exam #1

Wednesday, Oct. 8, 7:30 - 9:00 pm, Swords 359

The first exam covers all of Chapter 1 of the course text. You should go over homework problems, your class notes and the first lab project. I have also chosen some good practice problems from the Chapter 1 review. The answers to all of these review problems are posted on the Moodle page for the course. The exam will be designed to take one hour although you will have the full 90 minutes.

Note: You will be allowed a scientific calculator for the exam that does NOT have graphing capabilities. Please bring your own calculator.

Exam Review: We will review for the exam during Monday's class, Oct. 6th. 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. First-order ODE's -- analytic techniques: separation of variables, guess and test (method of undetermined coefficients), integrating factors, Linearity principles

  4. First-order ODE's -- numerical techniques: Euler's method, slope fields

  5. First-order ODE's -- qualitative techniques and concepts: 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. 136 - 141)
Problems:   1, 2, 4, 5, 7, 9, 11, 17, 19, 20, 21, 23, 25, 26, 29, 31, 33, 34, 35, 37, 39, 40, 41, 42, 46, 47, 48, 49, 50, 51, 52

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.