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EAMCET Engineering 2005 Solved Paper
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© examsnet.com
Question : 158
Total: 160
If
x
2
y
−
x
3
d
y
d
x
=
y
4
cos
x
, then
x
3
y
is equal to
s
i
n
x
2
s
i
n
x
+
c
−
3
s
i
n
x
+
c
3
cos
x
+
c
Validate
Solution:
Given that
x
2
y
−
x
3
d
y
d
x
=
y
4
cos
x
On dividing by
y
4
, we get
x
2
y
3
−
x
3
y
4
d
y
d
x
=
cos
x
⇒
x
3
y
4
d
y
d
x
=
x
2
y
3
−
cos
x
⇒
1
y
4
d
y
d
x
−
1
x
y
3
=
−
1
x
3
cos
x
Let
−
1
y
3
=
t
⇒
1
y
4
d
y
d
x
=
1
3
⋅
d
t
d
x
⇒
1
3
⋅
d
t
d
x
+
1
t
x
=
1
x
3
cos
x
⇒
d
t
d
x
+
3
x
t
=
3
x
3
cos
x
This is a linear differential equation in
t
, on comparing with
d
t
d
x
+
P
t
=
Q
, we get
P
=
3
x
,
Q
=
3
x
3
cos
x
∴
I.F.
=
e
∫
P
dx
=
e
∫
3
x
dx
=
e
log
x
3
=
x
3
∴
Complete solution is
t
x
3
=
3
∫
x
3
x
3
cos
x
dx
+
c
1
⇒
t
x
3
=
3
s
i
n
x
+
c
1
⇒
−
1
y
3
x
3
=
3
s
i
n
x
+
c
1
⇒
y
−
3
x
3
=
−
3
s
i
n
x
+
c
© examsnet.com
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