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66.
Use differential equation graphing mode.
For reference, the equations of the solution curves are as
follows.
(0, 1):
y
52}
x
2
1
1
}
(0, 2):
y
52}
2
x
2
2
1
}
(0,
2
1):
y
52}
x
1
1
1
}
(0, 0):
y
5
0
[
2
2.35, 2.35] by [
2
1.55, 1.55]
The concavity of each solution curve indicates the sign of
y
0
.
67. (a)
}
d
d
x
}
(ln
x
1
C
)
5 }
1
x
}
for
x
.
0
(b)
}
d
d
x
}
[ln (
2
x
)
1
C
]
5 }
2
1
x
} }
d
d
x
}
(
2
x
)
5
1
}
2
1
x
}
2
(
2
1)
5 }
1
x
}
for
x
,
0
(c)
For
x
.
0, ln
)
x
)
1
C
5
ln
x
1
C
, which is a solution
to the differential equation, as we showed in part (a).
For
x
,
0, ln
)
x
)
1
C
5
ln (
2
x
)
1
C
, which is a
solution to the differential equation, as we showed in
part (b). Thus,
}
d
d
x
}
ln
)
x
)
5 }
1
x
}
for all
x
except 0.
(d)
For
x
,
0, we have
y
5
ln (
2
x
)
1
C
2
, which is a
solution to the differential equation, as we showed in
part (a). For
x
.
0, we have
y
5
ln
x
1
C
1
, which is a
solution to the differential equation, as we showed
part (b). Thus,
}
d
d
y
x
} 5 }
1
x
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Spring '08 term at University of Florida.
 Spring '08
 GERMAN
 Equations, Mean Value Theorem

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