1.

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

Answer» Given: `y=sin^(-1)x." …(i)"`
On differentiating both sides of (i) w.r.t. x, we get
`y_(1)=(1)/(sqrt(1-x^(2))`
`rArr y_(1)^(2)=(1)/((1-x^(2)))`
`rArr(1-x^(2))y_(1)^(2)=1." …(ii)"`
On differentiating both sides of (ii) w.r.t. x, we get
`(1-x^(2)).2y_(1)y_(2)+y_(1)^(2)(-2x)=0`
`rArr (1-x^(2))y_(2)-xy_(1)=0.`
Hence, `(1-x^(2))(d^(2)y)/(dx^(2))-x(dy)/(dx)=0.`


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