Question

If y = `[x + sqrt(x^2 - 1)]^15 + [x - sqrt(x^2 - 1)]^15`, then `(x^2 - 1)(d^2y)/(dx^2) + x(dy)/(dx)` is equal to ______.

Options

• 12 y

• 224 y²

• 225 y²

• 225 y

Answer 1

If y = `[x + sqrt(x^2 - 1)]^15 + [x - sqrt(x^2 - 1)]^15`, then `(x^2 - 1)(d^2y)/(dx^2) + x(dy)/(dx)` is equal to 225 y.

Explanation:

y = `{x + sqrt(x^2 - 1)}^15 + {x - sqrt(x^2 - 1)}^15`

Differentiate w.r.t. 'x'

`(dy)/(dx) = 15(x + sqrt(x^2 - 1))^14 [1 + x/sqrt(x^2 - 1)] + 15(x - sqrt(x^2 - 1))^14(1 - x/sqrt(x^2 - 1))`

`\implies (dy)/(dx) = 15/sqrt(x^2 - 1).y`  ...(i)

`\implies sqrt(x^2 - 1). (dy)/(dx)` = 15 y

Again differentiating both sides w.r.t. x

`x/sqrt(x^2 - 1) . (dy)/(dx) + sqrt(x^2 - 1) (d^2y)/(dx^2) = 15(dy)/(dx)`

`\implies x(dy)/(dx) + (x^2 - 1)(d^2y)/(dx^2)`

= `15sqrt(x^2 - 1). 15/sqrt(x^2 - 1) .y`

= 255 y