What is the integrating factor of the differential equation `(1-y^2)(dx)/(dy)+y x=a y(-1

Options

A. `(1)/(y^(2)-1)`
B. `(1)/(sqrt(y^(2)-1))`
C. `(1)/(1-y^(2))`
D. `(1)/(sqrt(1-y^(2)))`

Explanation

Given differential equation
`(1-y^(2))(dx)/(dy)+yx=ay`,
`implies (dy)/(dx)+((y)/(1-y^(2)))x=(ay)/(1-y^(2))`
Here, `P=(y)/(1-y^(2))`, `Q=(ay)/(1-y^(2))`
`I.F.=e^(int(y)/(1-y^(2))dx)`
`impliesI.F.=e^(-(1)/(2)int(-2y)/(1-y^(2))dy)`
`implies I.F.=e^((-1)/(2)log(1-y^(2))`
`implies I.F.=e^(log(1-y^(2))^(-1//2)`
`=(1-y^(2))^(-1//2)=(1)/(sqrt(1-y^(2)))`