Q.) The general solution of
dy/dx=2x-y/x+2y is given by
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Answer:
Given
dx
dy
=
x+2y
2x−y
Put y=vx
dx
dy
=v+x
dx
dv
∴ v+x
dx
dv
=
1+2v
2−v
⇒ x
dx
dv
=
1+2v
2−v−v(1+2v)
⇒ x
dx
dv
=
1+2v
2−2v−2v
2
⇒ ∫
2(1−v−v
2
)
1+2v
=∫
x
1
dx
⇒ loga−
2
1
log1−v−v
2
)=logx
⇒ 2loga−log(1−v−v
2
=2logx
⇒ logc=log[x
2
(1−v−v
2
] [Let a
2
=c]
⇒ c=x
2
(1−v−v
2
)
⇒ c=x
2
(1−
x
y
−
x
2
y
2
)
⇒ c=x
2
(
x
2
x
2
−xy−y
2
)
⇒ x
2
−xy−y
2
=c
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