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Unformatted text preview: 26 1 Solutions 29. Let
denote the volume of ﬂuid in the container at time . Initially,
there are L. For each minute that passes there is a net gain of
L
of ﬂuid. So
. The container overﬂows when
or
minutes.
L
g
g
.
input rate: input rate
min
L
min
L
g
g
output rate: output rate
min
L
min
Since
input rate output rate, it follows that
satisﬁes the initial
value problem Simplifying and putting in standard form gives the equation ,
, and the
The coeﬃcient function is
integrating factor is
. Multiplying the standard form equation
by the integrating factor gives
. Integrating and
simplifying gives
, where is a constant. The initial
condition
implies
so
At the time the container overﬂows
we have
g of salt.
30. Let
denote the volume of ﬂuid in the tank at time . Initially, there
are
gallons of ﬂuid. For each minute that goes by there is a net increase
of
gallons. It follows that
. The tank will overﬂow
when
. Solving
gives
. Thus
minutes. Next we ﬁnd
:
gal
lbs
lbs
input rate: input rate
.
min
gal
min
gal
lbs
lbs
output rate: output rate
.
min
gal
min
Since
input rate output rate, it follows that
satisﬁes the initial
value problem Putting this equation in standard form gives The coeﬃcient function is
integrating factor is ,
. Thus , and the
. Integrating ...
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 Fall '08
 BELL,D

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