Exercises 3.2
find
x
(
t
). We can determine the concentration of salt in the first tank by dividing
x
(
t
) by the
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
=
x
(
t
)
20
kg
/
min
.
Since the incoming liquid is pure water, we conclude that
input rate
1
= 0
.
Therefore,
x
(
t
) satisfies 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 find
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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 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations, Constant of integration, Boundary value problem

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