Page 34 - Advanced engineering mathematics

14 CHAPTER 1 First-Order Differential Equations

SECTION 1.1 PROBLEMS

In each of Problems 1 through 6, determine whether 23. A thermometer is carried outside a house whose ambi-
y = ϕ(x) is a solution of the differential equation. C is ent temperature is 70 Fahrenheit. After five minutes,
◦
constant wherever it appears. the thermometer reads 60 , and fifteen minutes after
◦
this, it reads 50.4 . What is the outside temperature
◦
√

1. 2yy = 1;ϕ(x) = x − 1for x > 1 (which is assumed to be constant)?
2. y + y = 0;ϕ(x) = Ce −x 24. A radioactive element has a half-life of ln(2) weeks.

3
2y + e x C − e x If e tons are present at a given time, how much will

3. y =− for x > 0;ϕ(x) =
2x 2x be left three weeks later?
2xy √ C 25. The half-life of Uranium-238 is approximately

4. y = for x =± 2;ϕ(x) =
9
2
2 − x 2 x − 2 4.5(10 ) years. How much of a 10 kilogram block of
2
x − 3 U − 238 will be present one billion years from now?

5. xy = x − y;ϕ(x) = for x = 0
2x 26. Given that 12 grams of a radioactive element decays
6. y + y = 1;ϕ(x) = 1 + Ce −x to 9.1 grams in 4 minutes, what is the half-life of this

element?
In each of Problems 7 through 16, determine if the dif-
27. Evaluate
ferential equation is separable. If it is, find the general
∞

solution (perhaps implicitly defined) and also any singu- e −t 2 −9/t 2 dt.
lar solutions the equation might have. If it is not separable, 0
do not attempt a solution. Hint:Let
∞

−t 2 −(x/t) 2
7. 3y = 4x/y 2 I (x) = e dt.

0

8. y + xy = 0 Calculate I (x) and find a differential equation for

∞ −t 2 √

9. cos(y)y = sin(x + y) I (x). Use the standard integral 0 e dt = π/2to
determine I (0), and use this initial condition to solve
10. e x+y y = 3x

for I (x). Finally, evaluate I (3).
11. xy + y = y 2

28. (Draining a Hot Tub) Consider a cylindrical hot tub
2
(x + 1) − 2y with a 5-foot radius and a height of 4 feet placed on

12. y =
2 y one of its circular ends. Water is draining from the tub

13. x sin(y)y = cos(y) through a circular hole 5/8 inches in diameter in the
2
x 2y + 1 base of the tub.

14. y =
y x + 1 (a) With k = 0.6, determine the rate at which the
x
15. y + y = e − sin(y) depth of the water is changing. Here it is useful

to write
16. [cos(x + y) + sin(x − y)]y = cos(2x) dh dh dV dV/dt

= = .
In each of Problems 17 through 21, solve the initial value dt dV dt dV/dh
problem. (b) Calculate the time T required to drain the hot tub
if it is initially full. Hint:Onewaytodothisis
2
2
17. xy y = y + 1; y(3e ) = 2 to write

2
18. y = 3x (y + 2); y(2) = 8 0 dt

T = dh.
x
2
19. ln(y )y = 3x y; y(2) = e 3 H dh

(c) Determine how much longer it takes to drain the
20. 2yy = e x−y 2 ; y(4) =−2

lower half than the upper half of the tub. Hint:
21. yy = 2x sec(3y); y(2/3) = π/3 Use the integral of part (b) with different limits

22. An object having a temperature of 90 Fahrenheit is for each half.
◦
placed in an environment kept at 60 . Ten minutes 29. Calculate the time required to empty the hemispheri-
◦
◦
later the object has cooled to 88 . What will be the cal tank of Example 1.7 if the tank is inverted to lie
temperature of the object after it has been in this envi- on a flat cap across the open part of the hemisphere.
ronment for 20 minutes? How long will it take for the The drain hole is in this cap. Take k = 0.8asinthe
object to cool to 65 ? example.
◦
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October 14, 2010 14:9 THM/NEIL Page-14 27410_01_ch01_p01-42