If $\mathrm{y}={\sin }^{-1}\left(\mathrm{x}\right),$ then $\left(1-{\mathrm{x}}^2\right){\mathrm{y}}_2-{\mathrm{xy}}_1=$

The givne function is $y={\sin }^{-1}x$

Differentiate both sides 

$\frac{dy}{dx}=\frac{1}{\sqrt{1-x^2}}$

Hence, 

$y_1=\frac{1}{\sqrt{1-x^2}}$

$\sqrt{1-x^2}{y}_1=1$

Squaring on both sides 

$\left(1-x^2\right){y_1}^2=1$

Differentiate both sides 

$\left(1-x^2\right)2y_1{y}_2-2x{y_1}^2=0$

Cancel out the common factor $2y_1$ in each term

$\left(1-x^2\right)y_2-x{y}_1=0$