if x√ 1-y^2+y√ 1-x^2=a then show that d^2y/dx^2=-a/(1-x^2)^3/2
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Answer:
Answer
x
1−y
2
+y
1−x
2
=1
Differentiating throughout with respect to x we get
x
2
1−y
2
1
⋅(−2y)
dx
dy
+
1−y
2
+y
2
1−x
2
1
⋅(−2x)+
1−x
2
dx
dy
=0
⇒
1−y
2
−xy
dx
dy
+
1−y
2
+
1−x
2
(−xy)
+
1−x
2
dx
dy
=0
⇒(
1−y
2
−xy
+
1−x
2
)
dx
dy
=
1−x
2
dy
−
1−y
2
⇒(
1−y
2
−xy+
(1−x
2
)(1−y
2
)
)
dx
dy
=(
1−x
2
xy−
(1−x
2
)(1−y
2
)
)
⇒
1−y
2
−(xy−
(1−x
2
)(1−y
2
)
)
dx
dy
=
1−x
2
(xy−
(1−x
2
)(1−y
2
)
)
⇒
dx
dy
=−
1−x
2
1−y
2
Hence
dx
dy
=−
1−x
2
1−y
2
.
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