Que
(xy-2y2) dx- (x2-3xy) dy=0
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Answer:
The given equation can be written as
dx
dy
=
3xy
(x
2
+y
2
)
.... (1)
Put y=vx⇒
dx
dy
=v+x
dx
dv
, equation (1) becomes
v+x
dx
dv
=−(
3xvx
x
2
+v
2
x
2
)=−(
3x
2
v
x
2
(1+v
2
)
)
∴v+x
dx
dv
=−(
3v
1+v
2
)
∴x
dx
dv
=
3v
−1−v
2
−v=
3v
−1−v
2
−3v
2
=
3v
−1−4v
2
∴
8
3
∫
1+4v
2
8v
dv=−∫
x
dx
Integrating we have
8
3
log(1+4v
2
)=−logx+logc.
⇒3log(1+4v
2
)+8logx=logc
⇒(1+4v
2
)
3
⋅x
8
=c
⇒(1+4
x
2
y
2
)
3
⋅x
8
=c
⇒(x
2
+4y
2
)
3
x
2
=c.
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