Page 50 - Advanced_Engineering_Mathematics o'neil

30 CHAPTER 1 First-Order Differential Equations

SECTION 1.4 PROBLEMS

In each of Problems 1 through 14, find the general solu- 14. y + 2 y = 3 y 2

tion. These problems include all three types discussed in x x
this section. 15. Consider the differential equation

1 1 ax + by + c
2

1. y = y − y + 1 y = F
x 2 x dx + py +r
1 2
−4/3
2. y + = y in which a,b,c,d, p,and r are constants. Show that
x x 3
this equation is homogeneous if and only if c =r = 0.

3. y + xy = xy 2
Thus, suppose at least one of c and r is not zero.
x y Then this differential equation is called nearly homo-

4. y = +
y x geneous. Show that if ap − bd = 0 it is possible to
y choose constants h and k such that the transforma-

5. y = tion x = X + h, y = Y + k results in a homogeneous
x + y
equation.
1 1 4
2
6. y = y − y −
2x x x
In each of Problems 16 through 19, use the idea from
7. (x − 2y)y = 2x − y Problem 15 to solve the differential equation.

8. xy = x cos(y/x) + y

y − 3

1 1 16. y = x + y − 1

9. y + y = y −3/4
x x 4
3x − y − 9
2

2
10. x y = x + y 2 17. y =

x + y + 1
1 2
2
11. y =− y + y x + 2y + 7
x x 18. y =

−2x + y − 9
2
3

12. x y = x y − y 3
2x − 5y − 9

2
−x
13. y =−e y + y + e x 19. y = −4x + y + 9

1.5 Additional Applications
This section is devoted to some additional applications of first-order differential equations. We
will need Newton’s second law of motion, which states that the sum of the external forces acting
on an object is equal to the derivative (with respect to time) of the product of the mass and
the velocity. When the mass is constant, dm/dt = 0, and Newton’s law reduces to the familiar
F = ma in which a = dv/dt is the acceleration.
Terminal Velocity An object is falling under the influence of gravity in a medium such as water,
air, or oil. We want to analyze the motion.
Let v(t) be the velocity of the object at time t. Gravity pulls the object downward, while
the medium retards the downward motion. Experiment has shown that this retarding force is
proportional in magnitude to the square of the velocity. Let m be the mass of the object, g the
usual constant acceleration due to gravity, and α the constant of proportionality in the retarding
force of the medium. Choose downward as the positive direction (this is arbitrary). Let F be the
magnitude of the total external force acting on the object. By Newton’s law,
dv
2
F = mg − αv = m .
dt
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
October 14, 2010 14:9 THM/NEIL Page-30 27410_01_ch01_p01-42