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Byju's Answer
Standard XII
Mathematics
General Equation of Hyperbola
Show that the...
Question
Show that the family of curves for which
d
y
d
x
=
x
2
+
y
2
2
x
y
, is given by
x
2
-
y
2
=
C
x
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Solution
The given differential equation is
d
y
d
x
=
x
2
+
y
2
2
x
y
.....(1)
This is a homogeneous differential equation.
Putting y = vx and
d
y
d
x
=
v
+
x
d
v
d
x
in (1), we get
v
+
x
d
v
d
x
=
x
2
+
v
2
x
2
2
v
x
2
⇒
v
+
x
d
v
d
x
=
1
+
v
2
2
v
⇒
1
+
v
2
2
v
-
v
=
x
d
v
d
x
⇒
1
-
v
2
2
v
=
x
d
v
d
x
⇒
2
v
1
-
v
2
d
v
=
d
x
x
Integrating on both sides, we get
∫
2
v
1
-
v
2
d
v
=
∫
d
x
x
⇒
∫
-
2
v
1
-
v
2
d
v
=
-
∫
d
x
x
⇒
log
1
-
v
2
=
-
log
x
+
log
C
⇒
log
1
-
v
2
+
log
x
=
log
C
⇒
log
1
-
v
2
x
=
log
C
⇒
1
-
v
2
x
=
C
⇒
1
-
y
2
x
2
x
=
C
⇒
x
2
-
y
2
=
C
x
Thus, the family of curves for which
d
y
d
x
=
x
2
+
y
2
2
x
y
is given by
x
2
-
y
2
=
C
x
.
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2
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