The integrating factor of the linear equation (4x'y-6)dx+xdy=0 is
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Step-by-step explanation:
Explanation:
2
(
y
−
4
x
2
)
d
x
+
x
d
y
=
0
Which we can re-arrange as follows:
d
y
d
x
=
−
2
(
y
−
4
x
2
)
x
=
8
x
−
2
y
x
∴
d
y
d
x
+
2
y
x
=
8
x
..... [A]
We can use an integrating factor when we have a First Order Linear non-homogeneous Ordinary Differential Equation of the form;
d
y
d
x
+
P
(
x
)
y
=
Q
(
x
)
Then the integrating factor is given by;
I
=
e
∫
P
(
x
)
d
x
=
exp
(
∫
2
x
d
x
)
=
exp
(
2
ln
x
)
=
exp
(
ln
x
2
)
=
x
2
And if we multiply the DE [A] by this Integrating Factor,
I
, we will have a perfect product differential;
x
2
d
y
d
x
+
x
2
x
y
=
8
x
3
∴
d
d
x
(
x
2
y
)
=
8
x
3
Which we can now directly integrate to get:
x
2
y
=
∫
8
x
3
d
x
∴
x
2
y
=
2
x
4
+
c
∴
y
=
2
x
2
+
c
x
2
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