Question

Find the derivatives of the following:

If y = `(sin^-1 x)/sqrt(1 - x^2)`, show that (1 – x²)y₂ – 3xy₁ – y = 0

Answer 1

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

`y sqrt(1 - x^2) = sin^-1x`

Differentiating with respect to x

`y xx 1/2 (1 - x^2)^(1/2 - 1) (- 2x) + sqrt(1  x^2) y_1 = 1/sqrt(1 - x^2)`

`- xy(1 - x^2)^(- 1/2) + sqrt(1 - x^2) y_1 = 1/sqrt(1 - x^2)`

`(1 - x^2)^(1/2) [- xy(1 - x^2)^(- 1/2) + (1 - x^2)^(1/2) y_1]` = 1

`- xy (1 - x^2)^(- 1/2) xx (1 - x^2)^(1/2) + (1 - x^2)^(- 1/2)  * (1 - x^2)^(1/2) y_1` = 1

`- xy + (1 - x^2) y_1` = 1

Differentiating with respect to x, we get

`- x * y_1 + y(-1) + (1 - x^2)y_2 + y_1 (0 - 2x)` = 0

`- xy_1 - y + (1 - x^2)y_2 - 2xy_1` = 0

`(1 - x^2)y_2 - 3xy_1 - y` = 0