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