Unformatted text preview: 1 Solutions 25 27. Let
are denote the volume of ﬂuid in the tank at time . Initially, there
gal of brine. For each minute that passes there is a net decrease of
gal of brine. Thus
gal.
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 Put in standard form, this equation becomes The coeﬃcient function is
,
, and the integrating factor is
. Multiplying
the standard form equation by the integrating factor gives Integrating and simplifying gives
condition
implies and hence Of course, this formula is valid for
no ﬂuid and hence no salt in the tank.
28. input rate: input rate
output rate: g
L minutes there is g
min L
g
min
L
output rate, it follows that output rate Since
input rate
value problem L
min . After The initial
so g
.
min
satisﬁes the initial Put in standard form, this equation becomes
. The coeﬃcient
function is
,
, and the integrating factor
is
. Multiplying the standard form equation by the integrating
factor gives
. Integrating and simplifying gives
The initial condition
implies
so
After
minutes we have
g of salt.
The concentration is thus
g/L ...
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 Fall '08
 BELL,D
 Boundary value problem, input rate, output rate

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