The solution of the differential equation (1−x2)dydx−xy=1 is (where, x<1, x∈R and C is an arbitrary constant )

dydx−x1−x2y=11−x2 I.F. =e−∫x1−x2 dx=e12∫−2x1−x2 dx=e12ln(1−x2)=1−x2Hence, the solution of the differential equation isy1−x2=∫1−x21−x2dxy1−x2=∫11−x2dxy1−x2=sin−1(x)+c

The solution of the differential equation (1−x2)dydx−xy=1 is (where, x<1, x∈R and C is an arbitrary constant )

dydx−x1−x2y=11−x2 I.F. =e−∫x1−x2 dx=e12∫−2x1−x2 dx=e12ln(1−x2)=1−x2Hence, the solution of the differential equation isy1−x2=∫1−x21−x2dxy1−x2=∫11−x2dxy1−x2=sin−1(x)+c

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The solution of the differential equation $(1 - x^{2}) \frac{d y}{d x} - x y = 1$ is (where, $|x|$ <1, $x \in R$ and C is an arbitrary constant )

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$y (1 - x^{2}) = \tan^{- 1} x + C$

$y \sqrt{1 - x^{2}} = \tan^{- 1} x + C$

$y \sqrt{1 - x^{2}} = \sin^{- 1} (x) + C$

$y (1 - x^{2}) = \sin^{- 1} x + C$

answer is C.

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Detailed Solution

$\begin{array}{l} \frac{d y}{d x} - \frac{x}{1 - x^{2}} y = \frac{1}{1 - x^{2}} \\ I . F . = e^{- \int \frac{x}{1 - x^{2}} d x} \\ = {e^{\frac{1}{2}}}^{\int - \frac{2 x}{1 - x^{2}} d x} = e^{\frac{1}{2} \ln (1 - x^{2}) = \sqrt{1 - x^{2}}} \\ H e n c e, t h e s o l u t i o n o f t h e d i f f e r e n t i a l e q u a t i o n i s \\ y \sqrt{1 - x^{2}} = \int \frac{\sqrt{1 - x^{2}}}{1 - x^{2}} d x \\ y \sqrt{1 - x^{2}} = \int \frac{1}{\sqrt{1 - x^{2}}} d x \\ y \sqrt{1 - x^{2}} = \sin^{- 1} (x) + c \end{array}$