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Byju's Answer
Standard XI
Mathematics
Variable Separable Method
If the curve ...
Question
If the curve
y
=
y
(
x
)
satisfies the differential equation
y
−
x
d
y
d
x
=
a
(
y
2
+
d
y
d
x
)
and always passes through a fixed point
(
1
,
1
)
,
then the total number of possible values of
a
is
(Assume the constant of integration to be zero)
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Solution
y
−
x
d
y
d
x
=
a
(
y
2
+
d
y
d
x
)
⇒
y
−
a
y
2
=
(
x
+
a
)
d
y
d
x
⇒
∫
d
y
y
(
1
−
a
y
)
=
∫
d
x
x
+
a
⇒
∫
1
y
d
y
+
a
1
−
a
y
d
y
=
∫
d
x
x
+
a
⇒
ln
|
y
|
−
ln
|
1
−
a
y
|
=
ln
|
x
+
a
|
Putting
(
1
,
1
)
,
0
−
ln
|
1
−
a
|
=
ln
|
1
+
a
|
⇒
1
|
1
−
a
|
=
|
1
+
a
|
⇒
|
(
1
+
a
)
(
1
−
a
)
|
=
1
⇒
1
−
a
2
=
±
1
⇒
a
2
=
0
,
2
⇒
a
=
0
,
±
√
2
Hence, the total number of values of
a
is
3
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0
Similar questions
Q.
If the curve
y
=
y
(
x
)
satisfies the differential equation
y
−
x
d
y
d
x
=
a
(
y
2
+
d
y
d
x
)
and always passes through a fixed point
(
1
,
1
)
,
then the total number of possible values of
a
is
(Assume the constant of integration to be zero)
Q.
The curve
y
(
x
)
satisfying the differential equation
y
−
x
d
y
d
x
=
a
(
y
2
+
d
y
d
x
)
passes through
(
1
,
1
)
, then the possible of
a
is
(assuming the constant of integration to be zero)
Q.
If a curve
y
=
f
(
x
)
passes through the point
(
1
,
−
1
)
and satisfies the differential equation,
y
(
1
+
x
y
)
d
x
=
x
d
y
,
then
f
(
−
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2
)
is equal to :
Q.
Let
y
=
y
(
x
)
be a curve satisfying the differential equation
y
(
d
2
y
d
x
2
)
=
2
(
d
y
d
x
)
2
.
If the curve passes through
(
2
,
2
)
and
(
8
,
1
2
)
,
then the value of
y
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)
is
Q.
If a curve
y
=
f
(
x
)
passes through point
(
1
,
−
1
)
and satisfy the differential equation
y
(
1
+
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y
)
d
x
=
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d
y
,
then
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(
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