If `y=(x+sqrt(1+x^2))^n` then `(1+x^2)(d^2y)/(dx^2)+x(dy)/(dx)`

Options

A. `n^(2)y`
B. `-n^(2)y`
C. `-y`
D. `2x^(2)y`

Answer

A

Explanation

Given, `y=(x+sqrt(1+x^(2)))^(n)`
`implies(dy)/(dx)=n[x+sqrt(1+x^(2))]^(n-1)(1+(x)/(sqrt(x)+1))`
`=(n[x+sqrt(1+x^(2))]^(n))/(sqrt(1+x^(2)))`
`implies" "((dy)/(dx))^(2)(1+x^(2))=n^(2)y^(2)`
Again differentiating, we get
`2(dy)/(dx).(d^(2)y)/(dx^(2))(1+x^(2))+2x((dy)/(dx))^(2)=2n^(2)y(dy)/(dx)`
`implies" "(d^(2)y)/(dx^(2))(1+x^(2))+x(dy)/(dx)=n^(2)y`