Math 2A
Review for Exam 2: Chapters 3 & 4
Winter 2004
Chapter 3: 3.1 – 3.3
 Growth and Decay Equations
 Newton’s Law of Cooling
 Mixtures
 Logistic
 Simple System of Equations (Lotka-Volterra Predator-Prey Model)
Chapter 4: 4.1 – 4.4, 4.6 – 4.7
 Given a general solution to a DE, determine if there is a particular solution given certain
boundary conditions
 Know how to determine if a set of functions is a fundamental set (remember the Wronskian)
 Know what a homogeneous DE is
 Know what a nonhomogeneous DE is
 Be able to verify a particular solution to a DE
 Given a DE and one solution, be able to use reduction of order (the shortcut equation) to find a
second solution
 Homogeneous Linear DEs with constant coefficients: Know how to distinguish among the 3
cases in order to find the solution using the auxiliary equation
 Know how to use Undetermined Coefficients to solve a nonhomogeneous DE
 Know how to use Variation of Parameters in order to solve a nonhomogeneous DE
 Cauchy-Euler Equation: Recognize the DE and know how to solve it
Review your notes, your quizzes, your homework, and the worksheet for chapter 3.
Bring to the Exam:
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ONE 8 ½ inch by 11 inch page BOTH SIDES with notes written in YOUR HANDWRITING.
The integral table
Calculator
PENCILS and ERASER (DO NOT use a PEN to take this exam).
Some Review Problems:
1. The population of bacteria in a culture grows at a rate proportional to the number of bacteria present at
time t. After 3 hours, there are 400 bacteria present. After 10 hours there are 2000 bacteria present.
What was the initial number of bacteria? Answer: 201
2. A thermometer reading 70 degrees F is placed in an oven preheated to a constant temperature. Through
a glass window in the oven door, an observer records that the temperature reads 110 degrees F after ½
minute and 145 degrees F after 1 minute. How hot is the oven? Answer: 390 degrees.
3. Air containing 0.06% carbon dioxide is pumped into a room whose volume is 8000 cubic feet. The air
is pumped in a rate of 2000 cubic feet, and the circulated air is then pumped out at the same rate. If
there is an initial concentration of 0.2% carbon dioxide, determine the subsequent amount at 10
minutes? Answer: 5.7 cubic feet.
4. The logistic equation for a population P (in thousands) at time t of a certain species is given by
dP
 P (2  P ) . If the initial population is 3000 ( P(0)  3 ), what is the limiting population? If the
dt
initial population is 1000, what is the population when t 
1
? Can a population of 1000 ever decline to
2
500? Answers: 2000, 2280, no.
5. Find a member of the family y  c1  c2 x 2 on (, ) that is the solution to the boundary value problem
xy '' y '  0, y (0)  1, y '(1)  6 . Answer: y  1  3x 2 .
6. Given: y '' 25 y  0 ; y1  e5 x . Find a second solution. Answer: y  e5x .
7. Use the procedures in Chapter 4 to find the general solution of each differential equation:
A. y '' 2 y ' 2 y  0 Answer: y  c1e(1 3) x  c2e(1 3) x
x
 5 
 5 

x   c2 sin 
x )
B. 2 y '' 2 y ' 3 y  0 Answer: y  e 2 (c1 cos 
 2 
 2 
1
C. y '' 2 y ' y  x 2e x Answer: y  c1e x  c2 xe x  x 4e x
12
x
2e
D. y '' y  x  x Answer: y  c1e x  c2e  x  e xtan 1 (e x )  1  e  xtan 1 (e x )
e e
1
2
E. 6 x y '' 5 xy ' y  0 Answer: y  c1 x  c2 x
2

1
3
F. x 2 y '' xy ' y  x3 Answer: y  c1 x  c2 x ln( x) 
1 3
x
4
8. Solve the following differential equations subject to the indicated conditions:
13 x 5  x
1
e  e  x  sin( x)
4
4
2
1
1
1
1
B. y '' y  sec3 ( x), y (0)  1, y '(0) 
Answer: y  cos( x)  sin( x)  sec( x)
2
2
2
2
A. y '' y  x  sin( x), y (0)  2, y '(0)  3 Answer: y 
9. Graph the following system (at least 3 periods) using your calculator. x1 represent predators and
x2 represent prey. Analyze the graph using complete sentences. What are the periods? For 3
complete periods and for t  0 , where do the predators equal the prey?
x1 '(t )  2 x1  2 x1 x2 , x1 (0)  1
x2 '(t )  x1 x2  x2 , x2 (0)  3