If `y={x+sqrt(x^2+1)}^m` , show that `(x^2+1)y_2+x y_1-m^2\ y=0`

Options

A. `m^(2)y`
B. `my^(2)`
C. `m^(2)y^(2)`
D. none of these

Answer

A

Explanation

We have,
`y^(1//m)={x+sqrt(1+x^(2))}`
`impliesy={x+sqrt(1+x^(2))}^(m)`
`implies(dy)/(dx)=m{x+sqrt(1+x^(2))}^(m-1){1+(x)/(sqrt(x^(2)+1))}=m({xsqrt(1+x^(2))}^(m))/(sqrt(1+x^(2)))`
`implies" "(dy)/(dx)=(my)/(sqrt(1+x^(2)))`
`impliesy_(1)""^(2)(1+x^(2))=m^(2)y^(2)`
`implies2y_(1)y_(2)(1+x^(2))+2xy_(1)""^(2)=2m^(2)yy_(1)`
`impliesy_(2)(1+x^(2))+y_(1)=m^(2)y`