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If y=(tan^(-1)x)^2 , then prove that (1+...

If `y=(tan^(-1)x)^2` , then prove that `(1+x^2)^2\ y_2+2x\ (1+x^2)y_1=2` .

A

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B

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C

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D

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Text Solution

Verified by Experts

The correct Answer is:
C
Given , ` y = (tan^(-1) x)^(2)`
On differentiating both sides w.r.t.x, we get
`(dy)/(dx) = 2 tan ^(-1) xx (d)/(dx) (tan^(-1) x) = 2 (tan^(-1)x) (1)/(1 + x^(2))`
or ` (1 + x^(2) )y_(1) = 2 tan^(-1) x `
Again , differentiating both sides w.r.t.x, we get
`(1 + x^(2)) (dy_(1))/(dx) + y_(1) (d)/(dx) (1 + x^(2)) = (2)/(1 + x^(2))`
`rArr (1 + x^(2))y_(2)+ y_(1) (0 + 2x) = (2)/(1 + x^(2))`
` rArr (1 + x^(2))^(2) y_(2) + 2x (1 + x^(2))y_(1) = 2` .
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