BC 3
Differential Equations Practice
Name:
(1)
A population of bacteria grows at a rate proportional to the number present. After 2 hours, there
are 300 bacteria, and after 8 hours, there are 1800. Write the differential equation that models
this situation and find how many bacteria were present at time t = 0?
(2)
AP BC 1991: 6 (non-calculator) A certain rumor spreads through a community at the rate of
dy
 2y(1 y) , where y is the proportion of the population that has heard the rumor at time t.
dt
(a)
What proportion of the population has heard the rumor when it is spreading the fastest?
IMSA
(b)
If at time t = 0, ten percent of the people have heard the rumor, find y as a function of t.
(c)
At what time t is the rumor spreading the fastest?
Diff Eq Practice p.1
F’10
BC 3
Differential Equations Practice
Name:
3. When the valve at the bottom of a cylindrical tank is opened, the depth of the liquid in the tank drops at a
rate proportional to the square root of the depth of the liquid. If y(t) is the depth of the liquid t minutes after
the valve is opened, then this can be modeled by the differential equation
y(t )  k y(t )
a. Describe in words what the differential equation says about the change in the depth of the
liquid.
b. Does the level of the liquid drop faster when the tank is full or when the tank is half full?
Explain.
c. Suppose that for a certain tank y(0) = 9 feet, and y(20) = 4 feet. Solve for y(t), then sketch the
graph of y vs t.
y
x
d. Is this a realistic model? Explain.
IMSA
Diff Eq Practice p.2
F’10
BC 3
Differential Equations Practice
4.
Name:
Solve each IVP.
a.
z = 0.15z, z(0) = 5000
b.
y = –0.2(y – 68), y(0) = 202
b.
P = 0.3 P (200 – P), P(0) = 25
5.
Show that the function y = Ae3t + Be–t, where A and B are constants, is a solution to the differential
equation y – 2y – 3y = 0.
6.
Find real numbers m and n so that the functions y = e2x and y = e5x are both solutions to the
differential equation my + ny + y = 0.
IMSA
Diff Eq Practice p.3
F’10
BC 3
Differential Equations Practice
7.
8.
Name:
Coffee is cooling in a ceramic cup. The initial temperature of this coffee is 190° F,
and the room temperature is 32°F (because your room happens to be a tent you are
using on a camping trip). After 5 minutes, the temperature of the coffee is 130° F.
a.
Write the IVP that represents this situation. (No solving of equations to
determine any constants that might be present in your IVP!)
b.
Determine how long (to the nearest tenth of a minute) the coffee’s temperature
will stay above 98.6° F?
x
and y(2) = –3.
2y
Use Euler’s method with a step size of 0.5 to estimate the value of y(4).
Consider the IVP y  
a.
b.
IMSA BC 3
Solve this IVP and use your solution to determine the exact value of y(4).
Your solution should be of the form where y is a function of x. No TI-89 or
integral tables, please.
Asst. DE p.4
Fall ‘10
BC 3
Differential Equations Practice
9.
Find a general solution to each of the following differential equations. Each
solution should be of the form where y is a function of x. No TI-89 or integral
tables, please.
a.
10.
Name:
y  3 x 2 e  y
b.
y  1  y
In a state park in Kenya, the population of topi in the year 2000 was 150. One year
later, the topi population was 180. This state park can hold no more than 900 topi
at any one time. It is assumed that the topi population in the park is growing
logistically.
a.
Write the IVP that represents this situation. (No solving of equations to
determine any constants that might be present in your IVP!)
b.
Determine a logistic function that models the topi population.
c
What is the topi population when it is growing most rapidly?
d.
In what year is the topi population growing most rapidly?
IMSA BC 3
Asst. DE p.5
Fall ‘10