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If y=log\ [x+sqrt(x^2+1)] , prove that (...

If `y=log\ [x+sqrt(x^2+1)]` , prove that `(x^2+1)(d^2\ y)/(dx^2)+x(dy)/(dx)=0`

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Given, ` y = log (x + sqrt(x^(2) +1))`, differentiating , w.r.t x, we get
` y_(1) = 1/(x+sqrtx^(2)+1) { 1+ x/(sqrt(x^(2)+1))} = 1/(sqrt(x^(2) +1))`
` Rightarrow sqrt(x^(2) +1) y_(1) =1 `
Again differentiating both sides , w.r.t x, we obtain
` d/(dx) ( (x^(2) +1)^(1/2) y_(1)) = d/(dx) (1)`
`Rightarrow sqrt(x^(2)+1) d/(dx) (y_(1)) + y_(1) d/(dx) (x^(2)+1)^(1/2) =0`
` sqrt(x^(2)+1) (d^(2)y)/(dx^(2)) + 1/2 y_(1) 1/(sqrt(x^(2)+1)) (2x)=0`
` (x^(2)+1) (d^(2)y)/(dx^(2)) + x (dy)/(dx) =0, ` as desired
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