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

Answer

`1/sqrt(1-y^(2))`

Explanation

The given differential equation is `(1-y^(2))(dx)/(dy)+yx=ay`
or `(dx)/(dy)+(yx)/(1-y^(2))=(ay)/(1-y^(2))`
This is a linear differential equation of the form
`(dx)/(dy)+Py=Q`, where `P=y/(1-y^(2))` and `Q = (ay)/(1-y^(2))`
The integrating factor (I.F.) is given by the relation.
I.F. `=e^(int_(pdy))=e^(int(y))/(1-y^(2))dy`
`=e^(-1/2log(1-x^(2)))=e^(log[1/sqrt(1-y^(2))]`
`=1/sqrt(1-y^(2))`