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

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roshan-rox | Valedictorian

Posted October 2, 2013 at 5:31 AM via web

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If `y=(1/2)(sin^-1 x)^2` , show that

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

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jeew-m | College Teacher | (Level 1) Educator Emeritus

Posted October 2, 2013 at 5:42 AM (Answer #1)

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

By differentiation the above;

`dy/dx = 1/2xx2sin^(-1)x(1/sqrt(1-x^2))`

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

Raise the above to the power 2.

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

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

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

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

By differentiation the above;

`(1-x^2)xx2(dy/dx)(d^2y)/(dx^2)+(dy/dx)^2(-2x) = 2dy/dx`

`(1-x^2)(d^2y)/(dx^2)-xdy/dx = 1`

`(1-x^2)(d^2y)/(dx^2)-xdy/dx -1 = 0`

So the required answer is proved.

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