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JEE Main Math Class 12 Differential Equations Part 2 Questions
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© examsnet.com
Question : 100
Total: 124
Let
y
=
y
(
x
)
be the solution of the differential equation
x
d
y
−
y
d
x
=
√
(
x
2
−
y
2
)
d
x
,
x
≥
1
, with
y
(
1
)
=
0
. If the area bounded by the line
x
=
1
,
x
=
e
π
,
y
=
0
and
y
=
y
(
x
)
is
α
e
2
π
+
β
, then the value of
10
(
α
+
β
)
is equal to
[18 Mar 2021 Shift 2]
Your Answer:
Validate
Solution:
x
d
y
−
y
d
x
=
√
x
2
−
y
2
d
x
⇒
x
d
y
−
y
d
x
x
2
=
1
x
√
1
−
y
2
x
2
d
x
⇒
d
(
y
x
)
√
1
−
(
y
x
)
2
=
1
x
d
x
On integrating.
∫
1
√
1
−
(
y
x
)
2
⋅
d
(
y
/
x
)
=
∫
1
x
d
x
⇒
sin
−
1
(
y
/
x
)
=
log
|
x
|
+
C
Now, at
x
=
1
,
y
=
0
∴
C
=
0
Hence,
y
=
x
sin
(
log
x
)
∴
A
=
∫
1
e
x
x
sin
(
log
x
)
d
x
Put
x
=
e
t
⇒
d
x
=
e
t
d
t
∴
A
=
π
∫
0
e
2
t
sin
(
t
)
d
t
(
using
∫
e
a
x
⋅
sin
b
x
=
e
2
x
a
2
+
b
2
(
a
sin
b
x
−
b
cos
b
x
)
)
α
e
2
π
+
β
=
(
e
2
t
5
(
2
sin
t
−
cos
t
)
)
0
π
=
1
+
e
2
π
5
∴
α
=
1
5
,
β
=
1
5
∴
10
(
α
+
β
)
=
4
© examsnet.com
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