The integrating factor of differential euation `(1-x^(2))(dy)/(dx)-xy=1 "is"`

Options

A. `-x`
B. `(x)/(1+x^(2))`
C. `sqrt(1-x^(2))`
D. `(1)/(x)log(1-x^(2))`

Answer

C

Explanation

Given that, `" "(1-x^(2))(dy)/(dx)-xy=1`
`rArr" "(dy)/(dx)-(x)/(1-x^(2))y=(1)/(1-x^(2))`
which is a linear differential equation.
`therefore" "IF=e^(-int(x)/(1-x^(2))dx)`
Put `" "1-x^(2)=trArr-2xdx=dtrArrxdx=-(dt)/(2)`
Now, `" "IF=e^((1)/(2)int(dt)/(t))=e^((1)/(2)logt)=e^((1)/(2)log(1-x^(2)))=sqrt(1-x^(2))`