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

634 Chapter 9 Differential Equations
9 Review
concept check
1. (a) What is a differential equation?
(b) What is the order of a differential equation?
(c) What is an initial condition?
2. What can you say about the solutions of the equation
y9 − x 2 1 y 2 just by looking at the differential equation?
3. What is a direction field for the differential equation
y9 − Fsx, yd?
4. Explain how Euler’s method works.
5. What is a separable differential equation? How do you solve
it?
6. What is a first-order linear differential equation? How do you
solve it?
Answers to the Concept Check can be found on the back endpapers.
7. (a) Write a differential equation that expresses the law of
natural growth. What does it say in terms of relative growth
rate?
(b) Under what circumstances is this an appropriate model for
population growth?
(c) What are the solutions of this equation?
8. (a) Write the logistic differential equation.
(b) Under what circumstances is this an appropriate model for
population growth?
9. (a) Write Lotka-Volterra equations to model populations of
food-fish sFd and sharks sSd.
(b) What do these equations say about each population in the
absence of the other?
TRUE-FALSE QUIZ
Determine whether the statement is true or false. If it is true, explain
why. If it is false, explain why or give an example that disproves the
statement.
1. All solutions of the differential equation y9 − 21 2 y 4 are
decreasing functions.
2. The function f sxd − sln xdyx is a solution of the differential
equation x 2 y9 1 xy − 1.
3. The equation y9 − x 1 y is separable.
4. The equation y9 − 3y 2 2x 1 6xy 2 1 is separable.
5. The equation e x y9 − y is linear.
6. The equation y9 1 xy − e y is linear.
7. If y is the solution of the initial-value problem
dy
dt
then lim t l ` y − 5.
− 2yS1 2 y 5D ys0d − 1
exercises
1. (a) A direction field for the differential equation
y9 − ysy 2 2dsy 2 4d is shown. Sketch the graphs of the
solutions that satisfy the given initial conditions.
(i) ys0d − 20.3 (ii) ys0d − 1
(iii) ys0d − 3 (iv) ys0d − 4.3
(b) If the initial condition is ys0d − c, for what values of
c is lim t l ` ystd finite? What are the equilibrium solutions?
y
6
4
2
0 1 2 x
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