The final exam is CUMULATIVE, that is, it covers all the material from the first day of class onwards. Approximately 25 - 30% will cover material since the second exam. This is sections 5.1, 5.2 and 5.3 from the course text. You should go over homework problems, the midterm exams and your class notes. I have also chosen some good practice problems from the Chapter 5 review exercises. The answers to the odd numbered questions are in the back of the book while the answers to the evens are given below. For practice problems for the other chapters, see previous review sheets: Exam 1 (Chapter 1) , and Exam 2 (Chapters 2, 3 and 4). The exam will be designed to take two hours although you will have the full 3 hours to complete the exam.

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 on Thursday, Dec. 14th from 12:30 - 2:00 pm in Swords 302. Please come prepared with specific questions.

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

1. General ODE terminology: general solution, particular solution, initial-value problem, initial condition, Existence and Uniqueness Theorems
2. Types of ODE's: order of, autonomous, linear, separable, homogeneous, nonhomogeneous, periodic, nonlinear, etc.
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, competitive species
8. First-order Systems: equilibrium points, phase portraits, vector field, direction field, sketching component graphs from phase portrait, Euler's method, Existence and Uniqueness Theorem
9. Linear Algebra: linear independence, basis, trace, determinant, eigenvalues and eigenvectors, characteristic polynomial, null space of a matrix
10. Planar Linear Systems: linearity principle, eigenvalues and eigenvectors, stability type (source, sink, saddle, spiral sink, center, etc.), straight-line solutions, sketching phase portraits, form of general solution for three cases (real distinct, complex or repeated eigenvalues), trace-determinant plane, bifurcations
11. Harmonic Oscillators: physical setup (parameters m, b and k), converting to a system, solving homogeneous and non-homogeneous cases, extended linearity principle, classification (overdamped, underdamped, critically damped, undamped), long-term behavior, describing the motion, sketching phase portraits and component graphs, complexification, beats and resonance
12. Nonlinear planar systems: equilibrium points, linearization and classification of an equilibrium point, nullclines, sketching directions fields and phase portraits
13. Planar Hamiltonian systems: general theory, showing a system is Hamiltonian, finding a Hamiltonian function H(x,y), conserved quantities, drawing phase portraits