Ordinary Differential Equations

MATH 304, Final Exam

Thursday, Dec. 18, 11:30 - 2:00 pm, Swords 302

The final exam is CUMULATIVE, that is, it covers all the material from the first day of class onwards. Approximately a third of the exam will cover material since the second exam. This is Sections 4.1, 5.1, 5.2, and 5.3 from the course text.

You should go over homework problems, the midterm exams, the computer labs, and your class notes. I have also chosen some practice problems from Section 4.1 and the Chapter 5 Review Exercises, listed below. The answers to all of the review problems are posted on the Moodle page for the course. For practice problems for the other chapters, see previous review sheets: Exam 1 (Chapter 1), and Exam 2 (Chapters 2 and 3). The exam will be designed to take two hours although you will have the full two and a half hours to complete the exam.

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

The following concepts, definitions, theorems, 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. 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 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 vs. cooperative systems

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, sketching component graphs, finding the general solution for three cases (real distinct, complex, or repeated eigenvalues), solving an initial-value problem, trace-determinant plane, bifurcations

11. Harmonic Oscillators: physical setup (parameters m, b, and k), converting to a system, finding the general solution, finding a specific solution given an initial condition, solving homogeneous and non-homogeneous cases, extended linearity principle, classification (overdamped, underdamped, critically damped, undamped), long-term behavior, describing the motion of the mass-spring, sketching phase portraits and component graphs

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 (e.g., solutions travel on level curves of the Hamiltonian), showing a system is Hamiltonian, finding a Hamiltonian function H(x,y), conserved quantities

Some Practice Problems:

Section 4.1 (pp. 399 - 402)
Problems:   4, 7, 10, 18, 35, 42

Chapter 5 Exercises (pp. 555 - 557)
Problems:   1, 2, 3, 4, 5, 6, 11, 14, 15, 17, 25 (a, b, c), 27, 28a