Download Unit 6 Differential Equations Review Topics you should know: Slope

Unit 6 Differential Equations Review
Topics you should know:
 Slope Fields
 Euler’s Methods
 Solving Separable Differential Equations
 Exponential Functions; Growth and Decay Applications
 Writing, Interpreting, and Solving Logistic Models Expressed as Differential
Equations
 Solving First Order Linear Differential Equations
What you should be able to do:
 Find the general solution of a differential equation by separation of variables
 Find a particular solution using the initial condition to evaluate the constant of
integration
 Understand that the proposed solution of a differential equation is a function (not a
number) and if it and its derivative(s) are substituted into the given differential
equation, the resulting equation is true.
 Growth-decay problems
 Slope fields – draw a slope field by hand; identify the differential equation for a
given slope field; sketch a particular solution on a given slope field; interpret a slope
field
 Use Euler’s Method (table) to approximate a solution
 Use integration by parts or partial fractions (from Unit 5) to find the antiderivative
after separating the variables
 Understand the logistic growth model, its asymptotes, and meaning
There are 15 multiple choice review problems posted on the website! Review those!
More practice:
1. Given

a.
b.
c.
d.
e.
dy
 2xy and y = 2 when x = 0, the value of x when y = e is
dx
0.307
0.554
0.693
1.000
2.718
1
2. If

dy x
 and y(3) = 4, then
dx y
a. x2 – y2 = -7
b. x2 + y2 = 72
c. x2 – y2 = 7
d. x2 – y2 = 5
e. x2 – y2 = 72

3. In the figure below, PC is tangent to the graph of y  f (x) at point P. Points A and B

are on PC and points D and E are on the graph of y  f (x) . Which of the following
statements are true?
I.
II.
III.
Euler's method uses the y-coordinate of point A to approximate the y-coordinate of
the function f (x) at point E after one step.
Euler's method uses the y-coordinate of point B to approximate the zero of the
function f (x) at point D after two steps.
Euler's method overestimates the y-coordinate of the function f (x) near the point P
because f (x) is decreasing.
A. I only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III
dy k
 , k is a constant, and if y = 2 when x = 1 and y = 4 when x = e, then, when
dx x
x = 2, what does y equal?
a. 2
b. 4
c. ln 8
d. ln 2 + 2
e. ln 4 + 2
4. If

2
5. The solution curve of y’ = y that passes through the point (2, 3) is
a. y  e x  3
b. y  2x  5
c. y  0.406e x
d. y  e x  (e 2  3)

ex

e. y 
0.406


6. If you use Euler’s method and two steps with x  0.1 for the d.e. y’ = y, with initial
 value y(0) = 1, then, when x = 0.2, y is approximately
a.
b.
c.
d.
e.
1.100
1.210
1.331
1.464
none of these

7. Which of these statements about Euler’s method is (are) true?
I.
It can be used to estimate solutions of differential equations numerically
dy
 F(x, y) where F is
II.
It cannot be applied to an equation of the form
dx
defined implicitly
III.
It should not be used on an interval on which the function becomes
infinite.

a.
b.
c.
d.
e.
I only
II only
III only
I and III only
I, II, and III
8. A cup of coffee at temperature 180F is placed on a table in a room at 68F. The d.e.
dy
 0.11(y  68); y(0)  180 . After 10 minutes, the
for its temperature at time t is
dt
temperature in F of the coffee is
a. 96
b. 100

c. 105
d. 110
e. 115
3
9. Approximately how long does it take the temperature of the coffee in Question 43 to
drop to 75F?
a. 10 min
b. 15 min
c. 18 min
d. 20 min
e. 25 min
10. The concentration of a medication injected into the bloodstream drops at a rate
proportional to the existing concentration. If the factor of proportionality is 30%
per hour, in how many hours will the concentration be one-tenth of the initial
concentration?
a. 3
1
b. 4
3
2
c. 6
3
2
d. 7

3
e. none of these


11. The rate of change of the volume, V, of water in a tank with respect to time, t, is
directly proportional to the square root of the volume. Which of the following is a
differential equation that describes this relationship?



A. V (t)  k t
B. V (t)  k V
dV
k t
C.
dt
dV
k
D.

dt
V
dV
k V
E.
dt


4
Find the particular solution for each differential equation given the initial conditions
dy
 y 2;
dx
y(0) 
1
3

dP
 9P  0; P(0)  6
dt

Word Problems:
1. A population is growing at a rate that is directly proportional to the size of the
population. If the population is 2000 initially, and has grown to 5000 by the end of the
second day, how long will it take to grow to 10,000?
2. A population is growing at a rate that is three times the size of the population. If the
population is 12 after 3 hours, what was the size of the population initially?
3. A number of students infected by measles in a certain school is given by the formula
below where t is the number of days after students are first exposed to an infected student.
200
P(t) 
1 e 5.3t

a. Find P(0) and explain its meaning in the problem.
b. Find lim P(t) and explain its meaning in the problem.
t

5
4. A certain wild animal preserve can support no more than 250 lowland gorillas. 28
gorillas were known to be in the preserve in 1970. Assume that the rate of growth of the
population is given the differential equation below where t is measured in years:
dP
 0.0004P(250  P)
dt
Find a formula for the gorilla population in terms of t and use it to determine how long it
will take for the population to reach the carrying capacity of the preserve.

6