1.

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

Answer» Given: `y=(tan^(-1)x)^(2)." …(i)"`
On differentiating both sides of (i) w.r.t. x, we get
`y_(1)=2 tan^(-1)x.(1)/((1+x^(2))`
`rArr (1+x^(2))y_(1)=2 tan^(-1)x`
`rArr (1+x^(2))^(2)y_(1)^(2)=4(tan^(-1)x)^(2)" [on squaring both sides]"`
`rArr (1+x^(2))^(2)y_(1)^(2)-4y=0." ...(ii)"`
On differentiating both sides of (ii) w.r.t. x, we get
`(1+x^(2))^(2).2y_(1)y_(2)+y_(1)^(2).2(1+x^(2)).2x-4y_(1)=0`
`rArr (1+x^(2))^(2)y_(2)+2x(1+x^(2))y_(1)-2=0.`
Hence, `(1+x^(2))^(2)y_(2)+2x(1+x^(2))y_(1)=2.`


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