James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

chapter 9 Review 635
9–11 Solve the initial-value problem.
2. (a) Sketch a direction field for the differential equation
7. 2ye y 2 y9 − 2x 1 3sx 8. x 2 y9 2 y − 2x 3 e 21yx 19. One model for the spread of an epidemic is that the rate of
y9 − xyy. Then use it to sketch the four solutions that
satisfy the initial conditions ys0d − 1, ys0d − 21,
ys2d − 1, and ys22d − 1.
(b) Check your work in part (a) by solving the differential
equation explicitly. What type of curve is each solution
curve?
9. dr 1 2tr − r, rs0d − 5
dt
10. s1 1 cos xdy9 − s1 1 e 2y dsin x, ys0d − 0
11. xy9 2 y − x ln x, ys1d − 2
3. (a) A direction field for the differential equation
y9 − x 2 2 y 2 is shown. Sketch the solution of the
initial-value problem
y9 − x 2 2 y 2 ys0d − 1
; 12. Solve the initial-value problem y9 − 3x 2 e y , ys0d − 1, and
graph the solution.
Use your graph to estimate the value of ys0.3d.
13–14 Find the orthogonal trajectories of the family of curves.
y
13. y − ke x 14. y − e kx
3
15. (a) Write the solution of the initial-value problem
2
dP
1
− 0.1PS1 2
P Ps0d − 100
dt
2000D
_3 _2 _1 0 1 2 3 x
and use it to find the population when t − 20.
(b) When does the population reach 1200?
_1
_2
_3
16. (a) The population of the world was 6.1 billion in 2000 and
6.9 billion in 2010. Find an exponential model for these
data and use the model to predict the world population
in the year 2020.
(b) According to the model in part (a), when will the world
population exceed 10 billion?
(b) Use Euler’s method with step size 0.1 to estimate
ys0.3d, where ysxd is the solution of the initial-value
problem in part (a). Compare with your estimate from
part (a).
(c) On what lines are the centers of the horizontal line
segments of the direction field in part (a) located? What
happens when a solution curve crosses these lines?
(c) Use the data in part (a) to find a logistic model for the
population. Assume a carrying capacity of 20 billion. Then
use the logistic model to predict the population in 2020.
Compare with your prediction from the exponential model.
(d) According to the logistic model, when will the world population
exceed 10 billion? Compare with your prediction in
part (b).
17. The von Bertalanffy growth model is used to predict the length
4. (a) Use Euler’s method with step size 0.2 to estimate
Lstd of a fish over a period of time. If L` is the largest length
ys0.4d, where ysxd is the solution of the initial-value
for a species, then the hypothesis is that the rate of growth in
problem
length is proportional to L` 2 L, the length yet to be achieved.
y9 − 2xy 2 ys0d − 1
(a) Formulate and solve a differential equation to find an
expression for Lstd.
(b) Repeat part (a) with step size 0.1.
(c) Find the exact solution of the differential equation and
compare the value at 0.4 with the approximations in
parts (a) and (b).
5–8 Solve the differential equation.
(b) For the North Sea haddock it has been determined that
L` − 53 cm, Ls0d − 10 cm, and the constant of proportionality
is 0.2. What does the expression for Lstd become
with these data?
18. A tank contains 100 L of pure water. Brine that contains 0.1 kg
5. y9 − xe 2sin x 2 y cos x 6. dx
of salt per liter enters the tank at a rate of 10 Lymin. The solution
is kept thoroughly mixed and drains from the tank at the
dt − 1 2 t 1 x 2 tx
same rate. How much salt is in the tank after 6 minutes?
spread is jointly proportional to the number of infected people
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