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Joint Entrance Examination
Mathematics
Differential Equations
Question 274
Question 274 of 490
Medium
MCQ
Let
f
f
f
satisfy
f
′
(
x
)
=
7
−
3
4
f
(
x
)
x
f'(x)=7-\dfrac34\dfrac{f(x)}{x}
f
′
(
x
)
=
7
−
4
3
x
f
(
x
)
,
x
>
0
x>0
x
>
0
,
f
(
1
)
≠
4
f(1)\ne4
f
(
1
)
=
4
. Then
lim
x
→
0
+
x
f
(
1
x
)
\lim_{x\to0^+}xf\left(\dfrac1x\right)
lim
x
→
0
+
x
f
(
x
1
)
:
A
exists and equals
4
/
7
4/7
4/7
B
exists and equals 4
C
does not exist
D
exists and equals 0
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Solutions · 1
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Official solution
d
y
d
x
+
3
4
x
y
=
7
\dfrac{dy}{dx}+\dfrac{3}{4x}y=7
d
x
d
y
+
4
x
3
y
=
7
. I.F.
=
x
3
/
4
=x^{3/4}
=
x
3/4
.
y
=
4
x
+
C
x
−
3
/
4
y=4x+Cx^{-3/4}
y
=
4
x
+
C
x
−
3/4
.
f
(
1
/
x
)
=
4
/
x
+
C
x
3
/
4
f(1/x) = 4/x+Cx^{3/4}
f
(
1/
x
)
=
4/
x
+
C
x
3/4
, so
x
f
(
1
/
x
)
=
4
+
C
x
7
/
4
→
4
xf(1/x)=4+Cx^{7/4}\to4
x
f
(
1/
x
)
=
4
+
C
x
7/4
→
4
as
x
→
0
+
x\to0^+
x
→
0
+
.
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