1 The general solution to y^ prime =xy^ 2 has the form?
Answers
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0
Answer:
y
(
x
)
=
−
2
x
2
+
C
Step-by-step explanation:
Let's separate our variables, IE, have each side of the equation only in terms of one variable. This entails
d
y
y
2
=
x
d
x
Integrate each side:
∫
d
y
y
2
=
∫
x
d
x
−
1
y
=
1
2
x
2
+
C
Note that we would technically have constants of integration on both sides, but we moved them all over to the right and absorbed them into
C
.
Now, let's get an explicit solution with
y
as a function of
x
:
−
1
=
1
2
x
2
y
+
C
y
y
(
1
2
x
2
+
C
)
=
−
1
y
=
−
1
1
2
x
2
+
C
Let's get the fraction out of the denominator. It just looks messy.
y
=
−
1
x
2
+
2
C
2
Well,
2
C
=
C
,
as
2
multiplied by a constant just yields another constant.
y
=
−
1
x
2
+
C
2
Thus,
y
(
x
)
=
−
2
x
2
+
C
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