Math 3411–Spring 2009
Some review items for Exam 3
•THE DATE FOR EXAM 3 IS THURSDAY, APRIL 9
•The test will be given in class. You will have the entire class period to work on it.
•It is very important that you write neatly and present your answers logically. This helps me a great
deal when determining partial credit. Messy and unreadable work will not be graded.
•Questions will cover material from sections 5.1, 5.3–5.9.
•Calculators are allowed, but no books or notes are allowed. You may bring blank scratch paper if you
want.
•See the syllabus for an explanation of how the tests count toward your final grade.
•Note: There might be sample problems posted on Dr. Hollis’ webpage. Check here:
http://www.math.armstrong.edu/faculty/hollis/classes/DE/
1. Section 5.1 Introduction
•You should understand the physical meanings of all the quantities in equations
(3) and (5) (homogeneous and nonhomogeneous equations) on pages 95-96; you
don’t need to be able to derive them, but at least understand what each equation
describes, and that m= mass, k= spring constant, etc.
2. Section 5.3 Structure of Solutions
•You should know what it means for two functions to be linearly independent, and
be able to use the Wronskian (which we discussed in class) to determine whether
functions are independent or not.
3. Section 5.4/5.5 The Characteristic Equation
•Be able to find general solutions (or particular solutions to initial value problems)
for homogeneous equations by writing down and solving the characteristic equation.
Make sure that you know the forms of the solutions corresponding to the three cases:
real and distinct roots, real repeated root, and complex roots.
•Make sure you know how to do arithmetic with complex numbers, including finding
the absolute value of a complex number (see page 113) and complex conjugates.
•You should definitely know the Euler-DeMoivre formula on page 114 and the gen-
eralization in equation (3) on page 115.
4. Section 5.6 Nonhomogeneous Equations
•You should understand and be able to use operator form, and understand how to
apply exponential shift to solve equations.
•Make sure that you know the general procedure for solving nonhomogeneous dif-
ferential equations so that you can calculate solutions, and solve for constants in
the case of initial-value problems.
5. Section 5.7 Real Solutions from Complex Solutions
•Understand how the complex exponential can be used to find solutions to nonhomo-
geneous equations where the nonhomogeneous term involves sines and/or cosines.
All of the examples on pages 128–130 are very good.
