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If Y = ( Sin − 1 X ) 2 , Prove that ( 1 − X 2 ) D 2 Y D X 2 − X D Y D X − 2 = 0 . - Mathematics

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Sum

If y = `(sin^-1 x)^2,` prove that `(1-x^2) (d^2y)/dx^2 - x dy/dx -2 = 0.`

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Solution

Here,

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

⇒ `y_1 = 2 sin^-1 x 1/sqrt(1-x^2)`

⇒ `y_2 = 2/(1-x^2) + (2x sin^-1 x)/(1-x^2)^(3/2)`

⇒ `y_2 = 2/(1-x^2) + (2x sin^-1 x)/((1-x^2)sqrt(1-x^2)`

⇒ `y_2 = 2/(1-x^2) + (xy_1)/((1-x^2)`

⇒ `y_2 (1-x^2) = 2 + xy_1`

⇒ `y_2 (1-x^2) - xy_1 - 2 =0`

⇒  Therefore,  `(1 -x^2) (d^2y)/dx^2 - x dy/dx - 2 = 0`

Hence proved.

Concept: Derivatives of Inverse Trigonometric Functions
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