Explanation

(c)
Given, $\left( 1 - x^{2} \right)\frac{dy}{dx} - xy = 1$
$\Rightarrow \ \frac{dy}{dx} - \frac{x}{1 - x^{2}}y = \frac{1}{1 - x^{2}}$
This is a linear equation, comparing with the equation
$\ \frac{dy}{dx} + Py = Q$
$\Rightarrow P = - \frac{x}{1 - x^{2}},\ Q = \frac{1}{1 - x^{2}}$
$\therefore IF = e^{\int_{}^{}{P\ dx}} = e^{\int_{}^{}{\frac{- x}{1 - x^{2}}dx}}$
$\Rightarrow IF = e^{\frac{1}{2}\log{(1 - x^{2})}} = \sqrt{1 - x^{2}}$