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