The final exam is CUMULATIVE, that is, it covers all the material from the first day of class onwards. Approximately 35% will cover material since the second exam. This is sections 7.2, 7.3, 8.1, 8.2, 8.4, 8.5, 9.1 and 9.4 of the course text. In many cases, your class notes will be more useful than the text as we covered certain topics in greater detail in class. You should be sure to go over homework problems and past exams as well.

Note: No calculators will be allowed for this exam. The only numerical computations required will be doable by hand. The exam will be designed to take two hours (twice the length of a midterm) although you will have the full 3 hours.

Exam Review: We will review for the exam on Wednesday, Dec. 15th from 6:00 - 7:30 pm in Swords 359. Please come prepared with specific questions.

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

1. General ODE terminology and theory: general solution, particular solution, initial-value problem, initial condition, Existence and Uniqueness Theorem, Picard iterates, continuous dependence on initial conditions
2. 1st-order ODE's -- analytic techniques: separation of variables, integrating factors (for linear ODE)
3. 1st-order ODE's -- qualitative techniques: equilibrium solutions, phase lines, slope fields, bifurcations, bifurcation diagrams, periodic solutions, the Poincare map
4. Population models: unlimited growth model, logistic population model (with harvesting, periodic harvesting, etc.), carrying capacity
5. Planar Linear Systems X' = AX -- analytic techniques: eigenvalues and eigenvectors, finding the general solution, Jordan (canonical) form, finding the change of coordinates matrix to put A in canonical form
6. Planar Linear SystemsX' = AX -- qualitative techniques: sketching phase planes, direction fields, straight-line solutions, stability types (sink, source, saddle, etc.), classification, trace-determinant plane, bifurcations
7. Harmonic Oscillators: physical setup (parameters m, b and k), converting to a system, solving homogeneous and non-homogeneous cases, classification (overdamped, underdamped, critically damped, undamped), long-term behavior, phase portrait and component graphs
8. Higher Dimensional Linear Systems: solving X' = AX, Jordan form, generalized eigenvectors, phase portraits, finding the change of coordinates matrix to put A in canonical form
9. Non-linear planar systems: equilibrium points, linearization and classification of an equilibrium point, hyperbolic equilibrium points, nullclines, sketching directions fields and phase portraits, integrals (conserved quantities)
10. Planar Hamiltonian systems: general theory, showing a system is Hamiltonian, finding a Hamiltonian function H(x,y), sketching level curves of H, drawing phase portraits