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Byju's Answer
Standard XII
Mathematics
Higher Order Derivatives
Find the gene...
Question
Find the general solution for the following differential equation:
x
2
d
y
+
y
(
x
+
y
)
d
x
=
0
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Solution
x
2
d
y
=
−
y
(
x
+
y
)
d
x
⟹
d
y
d
x
=
−
y
(
x
+
y
)
x
2
d
y
d
x
=
−
x
y
x
2
−
y
2
x
2
d
y
d
x
=
−
y
x
−
(
y
x
)
2
The above differential equation is homogeneous.
∴
put
v
x
=
y
.
.
.
.
.
.
.
.
(
1
)
d
y
d
x
=
−
v
−
v
2
.
.
.
.
.
(
2
)
differentiating equation (1),
d
y
d
x
=
v
+
x
d
v
d
x
.
.
.
.
.
.
(
3
)
by equation (2) & (3),
v
+
x
d
v
d
x
=
−
v
2
−
v
x
d
v
d
x
=
−
v
2
−
2
v
⟹
1
v
2
+
2
v
d
v
=
−
1
x
d
x
integrating both sides,
∫
1
v
2
+
2
v
d
v
=
−
∫
1
x
d
x
∫
1
(
v
2
+
2
v
+
1
)
−
1
d
v
=
−
l
n
|
x
|
+
l
n
|
c
|
∫
1
(
v
+
1
)
2
−
1
2
d
v
=
−
l
n
|
x
|
+
l
n
|
c
|
using
∫
1
x
2
−
a
2
d
x
=
1
2
a
l
n
|
x
−
a
x
+
a
|
+
c
1
2
l
n
∣
∣
∣
v
+
1
−
1
v
+
1
+
1
∣
∣
∣
=
−
l
n
|
x
|
+
l
n
|
c
|
1
2
l
n
∣
∣
∣
v
v
+
2
∣
∣
∣
=
−
l
n
|
x
|
+
l
n
|
c
|
l
n
∣
∣
∣
√
v
√
v
+
2
∣
∣
∣
=
−
l
n
|
x
|
+
l
n
|
c
|
l
n
∣
∣
∣
√
v
√
v
+
2
∣
∣
∣
+
l
n
|
x
|
=
l
n
|
c
|
l
n
∣
∣
∣
x
√
v
√
v
+
2
∣
∣
∣
=
l
n
|
c
|
⟹
x
√
v
√
v
+
2
=
c
from equation (1)
v
=
y
x
,
x
√
y
x
√
y
x
+
2
=
c
squaring both sides,
x
y
(
y
+
2
x
x
)
=
c
2
Therefore,
general solution
:
x
2
y
=
c
2
(
y
+
2
x
)
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