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If `y = (tan^(-1))^(2)`, show that `(1+x^(2))(d^(2)y)/(dx^(2)) + 2x(1+x^(2))(dy)/(dx) - 2 = 0` |
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Answer» Given, ` y = (tan^(-1) x)^(2)` Let `tan^(-1) x = z` `x = tan z ` On differentiating both sides or ` dx = sec^(2) z dz` ` = (1 + tan^(2) z) dz` ` = (1 + tan^(2)z) dz` ` = (1 + x^(2)) dz` Now, ` y = z^(2)` or ` dy = 2z dz` ` = 2 tan^(-1) x dz` ` dy = 2 tan^(-1) x * (dx)/((1+x^(2))` Hence, ` (dy)/(dx) = (2 tan^(-1))/((1+x^(2)))` or `(1+x^(2))(dy)/(dx) = 2 tan^(-1) x` or `(1 +x^(2))^(2)((dy)/(dx))^(2) = 4 (tan^(-1)x)^(2)` or ` (1+x^(2))^(2)((dy)/(dx))^(2) - 4y = 0` Differentiating both sides with respect to x, `(1+x^(2))^(2)[2(dy)/(dx)] (d^(2)y)/(dx^(2)) + 2 ((dy)/(dx))^(2) (1+x^(2)) 2x - 4 (dy)/(dx)` = 0 or ` (1 + x^(2))^(2) (d^(2)y)/(dx^(2))+2x (1+x^(2))(dy)/(dx) - 2 = 0` |
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