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Exercises 3.2
fnd
x
(
t
). We can determine the concentration oF salt in the frst tank by dividing
x
(
t
)bythe
its volume, i.e.,
x
(
t
)
/
60 kg/gal. Note that the volume oF brine in this tank remains constant
because the ﬂow rate in is the same as the ﬂow rate out. Then
output rate
1
= (3 gal
/
min)
·
±
x
(
t
)
60
kg
/
gal
²
=
x
(
t
)
20
kg
/
min
.
Since the incoming liquid is pure water, we conclude that
input rate
1
=0
.
ThereFore,
x
(
t
) satisfes the initial value problem
dx
dt
= input rate
1
−
output rate
1
=
−
x
20
,x
(0) =
x
0
.
This equation is linear and separable. Solving and using the initial condition to evaluate the
arbitrary constant, we fnd
x
(
t
)=
x
0
e
−
t/
20
.
Now, let
y
(
t
) denote the mass oF salt in the second tank at time
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.
 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations

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