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

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

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We have, `y=(tan^(-1)x)^2`

Differentiating w.r.t. x

`y'=2 tan^(-1)x xx1/(1+x^2)`

`(1+x^2)y'=2 tan^(-1)x `

Again Differentiating w.r.t. x

`[y'(1+x^2)]'=2 xx1/(1+x^2) `

`[y'(1+x^2)]'=2 /(1+x^2) `

Using Product Rule

`(1+x^2)'y'+y''(1+x^2)=2 /(1+x^2) `

`2xy'+y''(1+x^2)=2/(1+x^2)`

`2xy'(1+x^2)+y''(1+x^2)xx(1+x^2)=2`

`2x(1+x^2)y'+y''(1+x^2)^2=2`

`y''(1+x^2)^2+2x(1+x^2)y'=2`

Hence Proved
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