Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Consider the two tanks shown in the figure below. Assume that tank A contains 50 gallons of water in which 25 pounds of
salt is dissolved. Suppose tank B contains 50 gallons of pure water. Liquid is pumped into and out of the tanks as indicated
in the figure; the mixture exchanged between the two tanks and the liquid pumped out of tank B are assumed to be well
stirred. We wish to construct a mathematical model that describes the number of pounds x₁(t) and x₂(t) of salt in tanks A
and B, respectively, at time t.
dx₁1
dt
dx1
dt
pure water
3 gal/min
mixture
4 gal/min
This system is described by the system of equations
1
50
2
25
dx2
dt
2
25
2
25*1
1
+
-X2
mixture
1 gal/min
-X2
B
with initial conditions x₁(0) = 25, x₂(0) = 0 (see (3) and the surrounding discussion on mixtures on page 107). What is the
system of differential equations if, instead of pure water, a brine solution containing 3 pounds of salt per gallon is pumped
into tank A?
-2
25*1 + 50x2+6×
dx2
2 - (25)*₁- (25)×₂2 -
=
x2
dt
mixture
3 gal/min
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Transcribed Image Text:Consider the two tanks shown in the figure below. Assume that tank A contains 50 gallons of water in which 25 pounds of salt is dissolved. Suppose tank B contains 50 gallons of pure water. Liquid is pumped into and out of the tanks as indicated in the figure; the mixture exchanged between the two tanks and the liquid pumped out of tank B are assumed to be well stirred. We wish to construct a mathematical model that describes the number of pounds x₁(t) and x₂(t) of salt in tanks A and B, respectively, at time t. dx₁1 dt dx1 dt pure water 3 gal/min mixture 4 gal/min This system is described by the system of equations 1 50 2 25 dx2 dt 2 25 2 25*1 1 + -X2 mixture 1 gal/min -X2 B with initial conditions x₁(0) = 25, x₂(0) = 0 (see (3) and the surrounding discussion on mixtures on page 107). What is the system of differential equations if, instead of pure water, a brine solution containing 3 pounds of salt per gallon is pumped into tank A? -2 25*1 + 50x2+6× dx2 2 - (25)*₁- (25)×₂2 - = x2 dt mixture 3 gal/min
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