`"If "y^(1//m)=(x+sqrt(1+x^(2)))," then "(1+x^(2))y_(2)+xy_(1)` is (where `y_(r)` represents the rth derivative of y w.r.t. x)

Options

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

Explanation

We have
`y^(1//m)=(x+sqrt(1+x^(2)))`
`"or "y=(x+sqrt(1+x^(2)))^(m)`
`"or "(dy)/(dx)=m(x+sqrt(1+x^(2)))^(m-1)(1+(x)/(sqrt(x^(2)+1)))`
`=m((x+sqrt(1+x^(2)))^(m))/(sqrt(1+x^(2)))`
`=(my)/(sqrt(1+x^(2)))`
`"or "y_(1)^(2)(1+x^(2))=m^(2)y^(2)`
`"or "2y_(1)y_(2)(1+x^(2))+2xy_(1)^(2)=2m^(2)yy_(1)`
`"or "y_(2)(1+x^(2))+xy_(1)=m^(2)y`