Exam #2 Review

Scheduling Information:

 Time: Wednesday, October 23, during your regular class time
Location: HH B103, our usual classroom.
Review Session: Monday, October 21, from 6:30-8:00pm in WEH 7500.
 

Reading:

 Section 4.1.
Section 4.3.
Section 4.4.
Section 5.1.1.
Section 5.1.2.
Section 5.1.3.
Section 7.1.
Section 7.2.
Section 7.3.
Section 7.4.1.

 

Review Questions:
  1. What is a particular solution to a second order linear non-homogeneous equation?
  2. What is the complementary solution to a second order linear non-homogeneous equation?
  3. What is the method of undetermined coefficients?
  4. What is "Rule 1" for the method of undetermined coefficients? "Rule 2?" "Rule 3?"
  5. How can you find the general solution to a second order, linear, constant-coefficeint, non-homogeneous eifferential equation?
  6. What is Hooke's law? What is a spring constant?
  7. What is the angular frequency of a mass-spring system? The period? The frequency?
  8. How can an undamped mass-spring system be modeled with a differential equation?
  9. What is the general solution for a mass spring system with no damping and no external forcing?
  10. What is the natural frequency of an undamped mass-spring system? The period? The frequency?
  11. What is the difference between an "angular frequency" and a "frequency"?
  12. What is the general solution for a mass-spring system with damping?
  13. What is overdamping? Underdamping? Critical damping? When will theseoccur?
  14. What is the quasi-frequency of an underdamped system? The quasi-period?
  15. What happens when an undamped mass-spring system is forced at a frequency different from its natural frequency?
  16. What happens when an undamped mass-spring system is forced at a frequency equal to its natural frequency?
  17. What is the Laplace transform? How can Laplace transforms be computed?
  18. What are the functions that can appear as solutions to Linear, Homogeneous, Constant-Coefficient, Second Order Differential Equations? What are the Laplace transforms of those functions?
  19. What is the Laplace transform L{f'(t)}? L{f''(t)}?
  20. How do you use Laplace transforms to solve second order linear equations?
  21. What is the Laplace transform L{exp(at)*f(t)}?
  22. What is the Laplace transform L{t*f(t)}?
  23. How can we determine inverse Laplace transforms? What technique from calculus class do we employ?
  24. How are Laplace transforms use to solve Linear, Homogeneous, Constant-Coefficient, Second Order Differential Equations?
  25. Explain the behavior of an undamped mass-spring system under the influence of a sinusoidal forcing function based on the Laplace transform. Why is the behavior different when the forcing frequency is the same/different from the natural frequency?
  26. How can you find the Laplace transform of a piecewise defined function?
 

Exercises:

 Section 4.1 #23, 27, 31, 33.
Section 4.3 #1, 5, 7, 13, 19, 23, 29, 35, 59, 61.
Section 4.4 #1, 5, 7, 19, 43.
Section 5.1 #1, 3, 5, 17, 19, 23, 33, 35.
Section 7.1 #1, 3, 5, 9, 11, 13, 19, 21, 31, 53, 55.
Section 7.2 #1, 5, 7, 13, 23, 27, 31, 33, 36, 46.
Section 7.3 #3, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 39, 41, 45, 47, 49, 51, 55, 57, 63, 67.
Section 7.4 #1, 3, 7, 9, 11, 17.
 

Old Exam Problems:

Here is a small collection of exam problems I've given to Differential Equations students in previous semesters. They are not all exactly like what we've done this semester, but most of them should still be good practice for you.

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.

 

Reference Tables:

I'll give you this table of Laplace transforms to use with your exam.