- (a) If y = (sin-1 x)2, then prove that (1-x²)y2-xy1-2=0. Hence show that (1-x²)yn+2-(2n+1)xyn+1-n yn=0 and find yn(0).
Answers
Answered by
0
Answer:
Given: y=(sin
−1
x)
2
To Prove: (1×x
2
)y
2
−xy
1
−2=0
Solution: Consider y=(sin
−1
x)
2
Differentiate 'y' with respect to 'x'
dx
dy
=2sin
−1
x.
1−x
2
1
Again differentiate
dx
dy
with respect to 'x'
dx
2
d
2
y
=2
⎣
⎢
⎢
⎢
⎡
(1−x
2
)
1−x
2
1
.
1−x
2
−sin
−1
x.
2
1−x
2
1
(−2x)
⎦
⎥
⎥
⎥
⎤
dx
2
d
2
y
=
(1−x
2
)
2[1+
1−x
2
x
sin
−1
x]
Now; LHS=(1−x
2
)y
2
−xy
1
−2
=(1−x
2
).2
⎣
⎢
⎢
⎡
(1−x
2
)
1+
1−x
2
x
sin
−1
x
⎦
⎥
⎥
⎤
−
1−x
2
x(2sin
−1
x)
+(−2)
=2+
1−x
2
2x
sin
−1
x−
1−x
2
x(2sin
−1
x)
−2
=0
=RHS
Hence proved
Similar questions