Page 347 - Advanced engineering mathematics

10.5 Applications and Illustrations of Techniques 327

Now use the initial condition to solve for the constants. Solve
⎛ ⎞ ⎛ ⎞
4 5c 2 + 10c 3 + 9
= c 1 + c 3 − 18/5 .
i(0+) = 4 ⎠ ⎝ ⎠
⎝
0 c 1 − 4c 2 − 9c 3 − 18/5
This can be written as
0 5 10 −5
⎛ ⎞ ⎛ ⎞
⎝ 1 0 1 ⎠ C = PC = 38/5 ⎠ .
⎝
1 −4 −9 18/5
Then
⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞
−5 1 4 5 5 −5 18/5
C = P −1 ⎝ 38/5 ⎠ = ⎝ 10 −10 10 ⎠⎝ 38/5 ⎠ = ⎝ −9 ⎠ .
10
18/5 −4 5 −5 18/5 4
The current is
⎛ −2t −20t/9 ⎞
9 − 4te + 40e
i = ⎝ 4e −20t/9 ⎠ .
36e −2t − 36e −20t/9

SECTION 10.5 PROBLEMS

1. Referring to the circuit of Figure 10.4, determine how pound of salt per gallon, tank 1 initially has 200
much time elapses between the time the switch is pounds of salt, and tank 2 initially has 150 pounds
closed and the time the charge on the capacitor is of salt. Determine the amount of salt in each tank at
a maximum. What is the maximum voltage on the time t > 0.
capacitor?
3. Two tanks are connected as shown in Figure 10.6.
2. Referring to Figure 10.5, tank 1 initially contains 200 Tank 1 initially contains 100 gallons of water in which
gallons of saltwater (brine), while tank 2 initially con- 40 pounds of salt are dissolved. Tank 2 initially con-
tains 300 gallons of brine. Beginning at time 0, brine tains 150 gallons of pure water. Beginning at t = 0, a
is pumped into tank 1 at the rate of 4 gallons per brine solution containing 1/5 pound of salt per gal-
minute, pure water is pumped into tank 2 at 6 gallons lon is pumped into tank 1 at the rate of 5 gallons
per minute, and the brine solutions are interchanged per minute. At this time, a solution which also con-
between the two tanks and also flow out of both tanks tains 1/5 pound of salt per gallon is pumped into
at the rates shown. The input to tank 1 contains 1/4 tank 2 at the rate of 10 gallons per minute. The

Brine 4 gal/min 12 gal/min Water 6 gal/min

Tank 1 Tank 2

6 gal/min

4 gal/min 12 gal/min
FIGURE 10.5 Connected tank system for Problem 2, Section 10.5.

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 20:32 THM/NEIL Page-327 27410_10_ch10_p295-342