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Question

lf y=(x2−1)n and yn denotes the nth derivative of y, then (x2−1)yn+2+2xyn+1=

A
(n2+1)yn
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B
(n21)yn
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C
n(n2+1)yn
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D
n(n+1)yn
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Solution

The correct option is D n(n+1)yn
y=(x21)n
y1=2xn(x21)n1
y2=2n(x21)n1+4x2(x21)n2n(n1)
(x21)y2=2n(x21)n+4x2(x21)n1n(n1)
(x21)y2=2ny+(n1)2xy1
(x21)y3+2xy2=2ny1+2(n1)y1+(n1)2xy2
(x21)y3=2(n+(n1))y1+(n2)2xy2
(x21)y4=2(n+(n1)+n2)y2+2(n3)xy3
.
.
(x21)yn+2=2(n+(n1)+(n2)+(n3)++(n(n1))+(nn))yn2xyn+1
(x21)yn+2=2(n+(n1)+(n2)+(n3)++1+0)yn2xyn+1
(x21)yn+2=n(n+1)yn2xyn+1
Hence, option D.

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