Differential Equations Exam 3 Study Guide
Differential Equations Exam 3 Study Guide
The exam will cover chapter 4 material (sections 4.1-4.4), Section 5.3, and chapter 6 sections 6.1-6.3. Typically the exams will be 5 questions in length with each question worth 20 points. Review class notes, and Section 4.1-4.4, Section 5.3, and Sections 6.1-6.3 outlines. Note some of the previous exam 2 and exam 3 questions you may not be able to do. If the material looks unfamiliar skip the question and move onto to the next. For the most recent years, old exams and solutions are provided. It would be good to go over previous exams 2 and 3 and look at the solutions in addition to doing the practice exam.
1. Two questions on the exam most likely will involve solving second order differential equations. Basic techniques of finding the characteristic equation and writing the general solution based on the roots of the characteristic equation are essential!! In addition you should be able to find the general solution of nonhomogeneous ODE with method of the undetermined coefficients (DON’T FORGET TO LOOK AT CASES WHERE method of undetermined coefficients fails and KNOW what adjustments to make to the initial guess of the particular solution)!!! It is also important to be familiar with knowing what terms of the basic harmonic oscillator equation describe the restoring force, the damping force, and external force. Know terminology such as undamped, damped, forced, unforced motion. (This information is found on top of p. 193).
Review HW
Section 4.1 p. 200-202 #3, 12, 13, 31, 36 (for extra practice #1-6, 13-16, 27-32)
Section 4.2 p. 213-215. #7, 10, 20, 28 (for extra practice #1-20)
Section 4.3 p. 225-226 #1, 6, 14, 19, 24 (for extra practice #1-16, 22-27)
Section 4.4 p. 236-237 #9, 16, 22, 25, 34 (for extra practice #1-36)
2. One question on the exam will involve eigenstuff (namely you need to be able to find eigenvalues and eigenvectors. Find the general solution of the linear system. Be able to check if the solutions are linearly independent.
Review HW
Section 5.3 p.301-303 #1, 4, 7, 9, 25, 27, 46 (for extra problems #1-18, 46-49)
3. One question may involve matching ODE vector fields and phase plane trajectories. Essentially, this type of problem requires use of your knowledge of eigenvalues and eigenvectors in drawing qualitative sketches and/or knowledge of nullclines to match the sketch. Be able to determine the equilibrium solution, sketch the horizontal and vertical nullclines, draw the direction of flow in the regions, and determine stability of the problem. (Be able to classify equilibrium points as a source, sink or saddle.
Review HW
Section 6.1 p. 331-333 problems #3, 8, 23-26 (for extra problems #1-10)
Review problems covered in lecture notes on November 11th and November 13th
*NOTE the old exams did not have a sample problem of drawing nullclines of linear systems because the material was covered differently! Suggestion. Make sure you can do the examples shown in lecture plus the problems #1-10 in Section 6.1.
See also Exam 2 Fall 2002 (Problem 4)
Exam 2 Spring 2001 (Problem 4)
Exam 3 Spring 2002 (Problem 1a)
Exam 3 Spring 2001 (Problem 2)
Exam 3 Spring 2000 (Problem 1b)
4. a) Part of a problem may involve defective matrix (i.e. a matrix with a double eigenvalue but with only has one eigenvector). You need to be able to find the generalized eigenvector. Be able to derive the condition that the generalized eigenvector must satisfy (namely [pic]).
Review HW
Sect 6.2 p.340 #15 (Practice #16), 28
b) Another portion of a problem may ask you to find the general solution with a non-defective matrix or the problem may request you to solve IVP.
Review HW
Sect 6.2 p. 340-341 #3, 6, 15, 10, 20, 42
Sect 6.3 p. 349-351 #5, 10, 13, 17 (any problems #1-16 for extra practice)
5. Some exams contain True/False Sections or Multiple Choice Sections that check your understanding of the material. Review your notes of the key points of each section and then try the following exam type questions.
Exam 2 (Spring 2000) Problem 3
Exam 3 (Fall 2001) Problem 5
Exam 3 (Spring 2001) Problem 4
Exam 3 (Fall 2000) Problem 4a, b, e
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