Integrating factor of the differential equation $\left(1-y^2\right) \frac{dx}{dy}+x y=a y$ is:

$\frac{1}{\sqrt{1-y^2}}$

The correct answer is Option (4) - $\frac{1}{\sqrt{1-y^2}}$

I.F. = ?

dividing eq. by $1-y^2$

so $\frac{dx}{dy}+\frac{y}{1-y^2}x=\frac{ay}{1-y^2}$

$I.F.=e^{\int\frac{y}{1-y^2}}dy=e^{-\frac{1}{2}\int\frac{-2y}{1-y^2}}dy$

$=e^{-\frac{1}{2}\log 1-y^2}=\frac{1}{\sqrt{1-y^2}}$