Exam #2 Review

Scheduling Information:

Time: Wednesday, March 22. 7:30 AM - 8:20 AM
Location: UC McConomy
Review Session: Tuesday, March 21, from 6:30-8:00pm in DH 2315.
 

Reading:

Section 7.1.
Section 7.2.
Section 7.3.
Section 7.4.1.
Section 4.1.
Section 5.1.

 

Review Questions:

  1. Why is p(t)y''+q(t)y'+r(t)y=g(t) called a *linear* equation?
  2. Why is it linear even if p, q, and r may not be linear functions of t?
  3. What can be said about linear combinations of solutions of homogeneous linear equations?
  4. What does it mean for a set of functions to be linearly independent?
  5. What is the fundamental set of solutions if the characteristic polynomial has distinct real roots? The general solution?
  6. What is the fundamental set of solutions if the characteristic polynomial has repeated real roots? The general solution?
  7. What is the fundamental set of solutions if the characteristic polynomial has complex roots? The general solution?
  8. How can you find solutions to initial value problems using the general solution?
  9. What is Hooke's law? What is a spring constant?
  10. What is the angular frequency of a mass-spring system? The period? The frequency?
  11. How can a damped mass-spring system be modeled with a differential equation?
  12. What is Hooke's law? What is a spring constant?
  13. What is the equation of motion (aka the "solution") for a damped mass spring system?
  14. What is the general solution for a mass spring system with no damping and no external forcing?
  15. What is the natural frequency of an undamped mass-spring system? The period? The frequency?
  16. What is the difference between an "angular frequency" and a "frequency"?
  17. What is the general solution for a mass-spring system with damping?
  18. What is the quasi-frequency? The quasi-period?
  19. What happens when an undamped mass-spring system is forced at a frequency different from its natural frequency?
  20. What happens when an undamped mass-spring system is forced at a frequency equal to its natural frequency?
  21. 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?
 

Exercises:

Section 7.1 #1, 3, 5, 7, 9, 11, 13, 19, 21, 31, 33, 41.
Section 7.2 #1, 5, 7, 13, 23, 31, 33, 36, 43.
Section 7.3 #3, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25.
Section 7.4 #1, 3, 7, 9, 11.
Section 4.1 #5, 7, 9, 15, 21, 23, 27.
Section 5.1 #1, 3, 9, 11.
 

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.