What is the integrating factor of the differential equation `(1-y^(2))(dx)/(dy)=ay(-1 lt ylt1)`?

Options

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

Answer

a

Explanation

The Given differential equation
` (1-y^(2)) (dx)/(dy) + yx = ay, -1 lt y lt 1 `
` rArr (dx)/(dy) + (y/(1-y^(2)))x = (ay)/(1-y^(2))`
[ on dividing by `(1-y)^(2)` both sides in the above equation ]
Here, ` P = y/(1-y^(2)), Q = (ay)/(1-y^(2))`
` :. IF = e^(int y/(1-y^(2))dy )rArrIF = e^(-1/2int(-2y)/(1-y^(2))dy)`
` rArr IF = e^(-1/2log (1-y^(2)))`
rArr IF = ` e^(log (1-y^(2))^(-1//2))`
` :. IF = = (1- y^(2))^(-1//2) = 1/(sqrt(1-Y^(2)))`