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Mathematics
Mathematics
Problem
Show that the one-parameter family
y
=
1
1
+
C
e
−
x
y=\dfrac{1}{1+Ce^{-x}}
y
=
1
+
C
e
−
x
1
satisfies the differential equation
d
y
d
x
=
y
(
1
−
y
)
\dfrac{dy}{dx}=y\,(1-y)
d
x
d
y
=
y
(
1
−
y
)
for all real
x
x
x
. Then determine the value of
C
C
C
that yields the solution satisfying
y
(
0
)
=
1
4
y(0)=\tfrac14
y
(
0
)
=
4
1
.
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