Question

If
`y = { \sin}^( - 1)x` then prove that
`( {1 - x) }^(2) \frac{ {d}^(2) y}{d {x}^(2) } - x (dy)/(dx) = 0`

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Answer 1

`{ \green{ \boxed{ \purple{ \sf{(1 - {x}^(2)) \frac{ {d}^(2)y }{d {x}^(2) } - x (dy)/(dx) = 0}}}}}`

Explanation:

Given,

`{ \blue{ \sf{y = { \sin}^( - 1)x}}}`

differentiate with respect to x

`{ \purple{ \sf{ (dy)/(dx) = \frac{1}{ \sqrt{1 - {x}^(2) } } }}}`

`{ \purple{ \sf{ \sqrt{1 - {x}^(2) } (dy)/(dx) = 1}}} \: { \to} \: { \tt{ {eq}^(n) (1)}}`

This is in the form of uv method

uv = uv' + vu' = 0

`{ \tt{ \sqrt{1 - {x}^(2) } = u}}`

`{ \tt{ (dy)/(dx) = v}}`

u' = differentiation of u

v' = differentiation of v

Apply uv method to Eqⁿ (1) then,

`{ \purple{ \sf{ \sqrt{1 - {x}^(2) } = \frac{ {d}^(2) y}{d {x}^(2) } + (dy)/(dx) \frac{1}{ \cancel2 \sqrt{1 - {x}^(2) } } (0 { \cancel{- 2}}x) = 0}}}`

Multiply by
`{ \red{ \sf{ \sqrt{1 - {x}^(2)}}}}` then,

`{ \boxed{ \purple{ \sf{(1 - {x}^(2)) \frac{ {d}^(2)y }{d {x}^(2) } - x (dy)/(dx) = 0}}}}`