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Solutions 1.5Page 51
Problem 27
Solve the differential equation by regarding y as the independent variable rather than x.
1
)
(
=
+
dx
dy
ye
x
y
The differential equation does not yet follow the general form given on pg.43.
The
derivative term (
dx
dy
) is differentiating y with respect to x.
Since y is the independent
variable, it must be the other way around.
Division by
dx
dy
, and rearranging terms yields
y
ye
x
dy
dx
=
−
dy
e
y
−
∫
=
1
)
(
ρ
.
This equation follows the form given on pg.43.
The integrating factor is
y
e
−
=
Multiplying both sides by
)
(
y
gives
y
x
e
dy
dx
e
y
y
=
−
−
−
The left hand side is
dy
x
e
d
y
)
(
−
, so the equation becomes
y
dy
x
e
d
y
=
−
)
(
Integrating both sides gives
C
y
x
e
ydy
dy
dy
x
e
d
y
y
+
=
=
−
−
∫
∫
2
)
(
2
Dividing by
gives
y
e
−
+
=
C
y
e
y
x
y
2
)
(
2
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View Full DocumentProblem 33
A tank contains 1000 liters (L) of a solution consisting of 100 kg of salt dissolved in
water.
Pure water is pumped into the tank at the rate of 5 L/s, and the mixture  kept
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 Spring '11
 HAFTKA

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