Exam #3 Review

Scheduling Information:

 Time: Monday, November 25, during your regular class time.
Location: HH B103, our regular class room.
Review Session: Saturday, November 23, from 10:00-11:30am in WEH 7500.
 

Topics:

 Section 7.5.
Section 3.3
Appendix II
Section 8.1
Section 8.2
Section 10.1
Section 10.2

 

Review Quesionts:
  1. What is the Dirac delta function?
  2. Why is the Dirac delta fiction not really a function?
  3. What is \int_{-\infty}^\infty f(t)\delta(t-c)dt?
  4. What is a system of differential equations? What is a solution to a system of differential equations?
  5. Under what conditions can you be sure that a system of differential equations will have unique solutions to initial value problems?
  6. What is a parametric curve? How are solutions to systems of differential equations related to parametric curves?
  7. What is an autonomous system of differential equations?
  8. What is the phase plane for a 2 dimensional autonomous system? How is the phase plane like the phase line of an autonomous differential equation?
  9. What are the nullclines of an autonomous system?
  10. How can nullclines be used to determine information about the direction of solutions to a 2-dimensional autonomous system? (along the nullclines [up/down or left/right] and in the regions between the nullclines [up-right, up-left, down-left, down-right])
  11. When is a system of differential equations "linear"?
  12. When is a system of differential equations "homogeneous"?
  13. When is a system of differential equations "constant coefficient"?
  14. What is the phase plane for a 2-dimensional system of differential equations?
  15. What is the dimension of a system of equations?
  16. What are eigenvalues of a matrix? How do you find them?
  17. What are eigenvectors of a matrix? How do you find them? What vector is never an eigenvector?
  18. What is the method for finding straight line solutions for a linear, homogeneous, constant coefficient system?
  19. When the coefficient matrix of a system has distinct real eigenvalues, how do you find the general solution?
  20. What is an "initial value problem" for a system of differential equations?
  21. How do you solve an initial value problem?
  22. When the coefficient matrix of a system has repeated real eigenvalues, how do you find the general solution?
  23. How do you find the general solution for a 2D system of linear homogeneous constant coefficient Differential equations when the eigenvalues are complex?
  24. How is the method for finding solutions for 2 dimensional systems of equations extended to find solutions for 3 dimensional systems?
  25. What is the phase plane for a 2-dimensional system of differential equations?
  26. How can you use analytic and qualitative methods to sketch trajectories for a 2-dimensional, first order, constant coefficient system of equations when the roots of the characteristic polynomial are complex? Distinct real? Repeated real?
  27. What is a "phase portrait" for a 2-dimensional, first order, linear system of equations?
  28. How can you use analytic and qualitative methods to sketch trajectories for a 2-dimensional, first order, constant coefficient system of equations when the roots of the characteristic polynomial are complex? Distinct real? Repeated real?
  29. What is a "phase portrait" for a 2-dimensional, first order, linear system of equations?
  30. What do the following terms mean when applied to 2-dimensional linear systems of differential equations: spiral, center, node, saddle, sink, source, improper, stable, asymptotically stable, unstable?
 

Exercises:

 Section 7.5 #1, 3, 5, 11, 15.
Section 3.3 #5, 7, 9, 11, 17 [Use nullclines to sketch solutions in the phase plane.]
Appendix II #47, 49, 53.
Section 8.2 #1, 3, 5, 9, 19, 21, 27, 33, 35, 37, 39, 49. [In addition to "find[ing] the general solution" you should (a) plot the nullclines and indicate where solutions are travelling left or right, up or down, or up-right, up-left, down-left, down-right, (b) on a separate set of axis sketch a phase portraait for the system (you can include the nullclines in this diagram, too), (c) classify the equilibrioum point (type and stability).]

 

Old Exam Problems:

Here is a collection of exam problems I've given to differential equations students in previous semesters.

And here is the table of Laplace transforms you will be given on the exam. The numbers on the table correspond to the entries in Table 6.2.1.