Final Exam Review

Be sure to review the material listed in the review pages for the First Exam, Second Exam, Third Exam, as well as the topics presented here. The Final Exam will be a cumulative exam.

Scheduling Information:

Time: Tuesday, May 9, from 1:00-4:00pm.
Location: DH 2210.
Review Session: Sunday, May 7, from 7:30-9:00pm, in MM A14.

Classification of Equilibrium Points.
Section 10.2.
Section 10.3.
Section 11.1.
Section 11.2.
Section 11.3.
Section 12.1.
Section 12.2.
Section 12.4

Review Questions:

1. What does it mean to say that T is a "period" of a function f?
2. What does it mean to say that T is the "fundamental period" of f?
3. What are the Fourier coefficients of a piecewise continuous function with period 2L?
4. What is the Fourier series of a piecewise continuous function with period 2L?
5. What is an even function? An odd function?
6. What useful facts do you know about integrals of even functions? Odd functions?
7. If f:[0,L]->R, how can you extend this to an even function on [-L,L]? On odd function on [-L,L]?
8. What is the Fourier sine series of a function f:[0,L]-R?
9. What is the Fourier cosine series of a function f:[0,L]->R?
10. What is a partial differential equation?
11. What is the wave equation?
12. What do the boundary conditions u(0,t)=u(L,t)=0 say about our vibrating string?
13. What does the initial condition u(x,0)=f(x) say about our vibrating string?
14. What does the initial condition du/dt(x,0)=g(x) say about our vibrating string?
15. What is separation of variables? What do we assume about out solutions u(x,t)?
16. How does the the separation of variables lead to the boundary value problem X"+\lambda X=0, X(0)=X(L)=0?
17. What are the solutions to the boundary value problem X"+\lambda X=0, X(0)=X(L)=0?
18. How do we solve our "wave equation problem" when g(x)=0? When f(x)=0? When both f(x) and g(x) are non-zero?

Exercises:

Section 10.3 #11, 13, 15, 29 (You should be able to sketch a phase portrait for these systems)
Section 11.2 #1, 5, 7, 21
Section 11.3 #1, 3, 5, 7, 11, 13, 17, 25, 27, 49.
Section 12.1 #1, 5, 7.
Section 12.4 #1, 3, 5.

Old Exam Problems:

Here is a final exam I gave when I taught Differential Equations in a previous semester. Also here is the table of Laplace transforms you will be given on the exam.

Before the questions come in - no, I don't have solutions available for these problems. You have a large number of problems from the text with answers, and it's important for you to get used to working on problems where the answers are not available. You'll have a chance to ask questions at the Review Session, and during your TA's office hours, too.