Question

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

Answer 1

Here, 

`log y = tan <sup>-1</sup> x `

Differentiating w.r.t. x, we get

`1/y. dy/dx = 1/(1+x^2)`

⇒`(1+x^2) dy/dx = y`

⇒ `(1+x^2) (d^2y)/dx^2 +2xdy/dx = dy /dx `

⇒ `(1+x^2) (d^2y)/dx^2 +2xdy/dx - dy /dx = 0 `

⇒ `(1+x^2) (d^2y)/dx^2 + (2x - 1) dy /dx = 0`

Hence proved.