If `y = (tan^(-1))^(2)`, show that
`(1+x^(2))(d^(2)y)/(dx^(2)) + 2x(1+x^(2))(dy)/(dx) - 2 = 0`

Explanation

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`