Page 132 - Advanced engineering mathematics

112 CHAPTER 3 The Laplace Transform

2 gal/min 3 gal/min

1/6 lb/gal

1 2

5 gal/min 2 gal/min

FIGURE 3.33 System of tanks in Problem 20, Section 3.6.

3 gal/min 3 gal/min

1 2

2 gal/min 4 gal/min 1 gal/min

FIGURE 3.34 System of tanks in Problem 21, Section 3.6.

19. Solve for the currents in the circuit of Figure 3.32 if Determine the amount of salt in each tank for any
E(t) = 5δ(t − 1). time t ≥ 0.
20. Two tanks are connected by a series of pipes as shown
21. Two tanks are connected by a series of pipes as shown
in Figure 3.33. Tank 1 initially contains 60 gallons
in Figure 3.34. Tank 1 initially contains 200 gallons of
of brine in which 11 pounds of salt are dissolved.
brine in which 10 pounds of salt are dissolved. Tank
Tank 2 initially contains 7 pounds of salt dissolved
2 initially contains 5 pounds of salt dissolved in 100
in 18 gallons of brine. Beginning at time zero, a mix-
gallons of water. Beginning at time zero, pure water is
ture containing 1/6 pound of salt for each gallon of
pumped into tank 1 at the rate of 3 gallons per minute,
water is pumped into tank 1 at the rate of 2 gallons
while brine solutions are interchanged between the
per minute, while salt water solutions are interchanged
tanks at the rates shown in the diagram. Three min-
between the two tanks and also flow out of tank 2 at
utes after time zero, 5 pounds of salt are dumped into
the rates shown in the diagram. Four minutes after
tank 2. Determine the amount of salt in each tank for
time zero, salt is poured into tank 2 at the rate of
any time t ≥ 0.
11 pounds per minute for a period of 2 minutes.
3.7 Polynomial Coefficients
3.7.1 Differential Equations with Polynomial Coefficients
If a differential equation has polynomial coefficients, we can use the Laplace transform if we
n
know how to take the transform of a function of the form t f (t). Begin with the case n = 1.

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October 14, 2010 14:14 THM/NEIL Page-112 27410_03_ch03_p77-120