Question

If y = `(x sin^-1 x)/sqrt(1 -x^2)`, prove that: `(1 - x^2)dy/dx = x + y/x`

Answer 1

Here, we have  y = `(x sin^-1 x)/sqrt(1 -x^2)`

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

Differentiate both sides w.r.t. x, we have

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

`- xy + (1 - x^2) (dy)/(dx) = x + sqrt(1 - x^2) sin^-1  x`

`- xy + (1 - x^2) (dy)/(dx) = x + sqrt(1 - x^2) . (y)/(x) sqrt(1 -x^2)    ...[ ∵ sin^-1 x = (y)/(x) sqrt(1 - x^2) , "using" (i) ]`

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

`- xy + (1 - x^2) (dy)/(dx) = x + (y)/(x) - yx`

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