Math 3411–Spring 2009
Some review items for Exam 2
•THE DATE FOR EXAM 2 IS TUESDAY, MARCH 3
•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 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 4.1, 4.2.
•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 3.1 Separable Equations
•Be able to use separation of variables, when applicable.
•Be able to use the substitution z=y/x when applicable.
2. Section 3.2 Toricelli’s Law
•You should memorize that Toricelli’s Law is
a(y)dy
dt =−ρ√y
where a(y) is the cross sectional area of the tank at height y, and ρis a constant.
You’ll need to be able to answer questions about tanks of varying shapes, and
questions about emptying time.
3. Section 3.3 Bernoulli and Riccati Equations
•You should be able to solve Bernoulli equations by making the substitution y=um
and finding an appropriate value for mthat renders the problem solvable.
•You don’t need to study Riccati equations.
4. Section 3.4 The Logistic Population Model
•You should be able to recognize (1) on page 63 as the differential equation that
describes logistic growth, and that (2) is its solution. It’s a good idea to be able to
solve (1); consider trying it as a Bernoulli equation.
•Harvesting: You don’t have to memorize (3) on page 65 or the formulas that follow
it, but you should know how to use them to answer questions similar to those in
the section.
5. Section 3.5 Direction Fields
•Given a differential equation in the form y0=f(t, y) you should be able to draw a
direction field, find isoclines, and draw sample solutions; see examples 3 and 4 on
page 69.
