The integrating factor of the differential equation $(1 - y^2) \frac{dx}{dy} + yx = ay, (-1 < y < 1)$ is:

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

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

The given differential equation is

$(1 - y^2) \frac{dx}{dy} + yx = ay, (-1 < y < 1)$

Rewrite the differential equation:

$\frac{dx}{dy} + \frac{yx}{(1-y^2)} = \frac{ay}{(1-y^2)}$

The integrating factor is $e^{\int \frac{y}{1-y^2} dy}$

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

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