Let a solution y=y(x) of the differential equation,1-x2dy-1-y2dx=0 satisfy y12=1.Statement-1 : y(x)=sinsin-1x+π3.Statement-2 : y(x) is given by y=x+3+3x22.

Given 1-x2dy-1-y2dx=0,⇒dydx=1-y21-x2 ⇒ ∫dy1-y2 = ∫dx1-x2∴ ⇒sin-1y=sin-1x+C …….(1)As y12=1 ⇒ sin-1(1)=sin-112+C ⇒C=π3.∴ From (1), we gety=sinsin-1x+π3 ⇒S-1is true.Also, y=x2+3-3x22 ⇒ S-2 is false.

Let a solution y=y(x) of the differential equation,1-x2dy-1-y2dx=0 satisfy y12=1.Statement-1 : y(x)=sinsin-1x+π3.Statement-2 : y(x) is given by y=x+3+3x22.

Given 1-x2dy-1-y2dx=0,⇒dydx=1-y21-x2 ⇒ ∫dy1-y2 = ∫dx1-x2∴ ⇒sin-1y=sin-1x+C …….(1)As y12=1 ⇒ sin-1(1)=sin-112+C ⇒C=π3.∴ From (1), we gety=sinsin-1x+π3 ⇒S-1is true.Also, y=x2+3-3x22 ⇒ S-2 is false.

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Let a solution $y = y (x)$ of the differential equation, $\sqrt{1 - x^{2}} d y - \sqrt{1 - y^{2}} d x = 0$ satisfy $y (\frac{1}{2}) = 1$ .
Statement-1 : $y (x) = \sin (\sin^{- 1} x + \frac{π}{3})$ .
Statement-2 : $y (x)$ is given by $y = \frac{x + \sqrt{3 + 3 x^{2}}}{2}$ .

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Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.

Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.

Statement-1 is true, statement-2 is false.

Statement-1 is false, statement-2 is true.

answer is C.

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Detailed Solution

Given $\sqrt{1 - x^{2}} d y - \sqrt{1 - y^{2}} d x = 0$

, $\Rightarrow \frac{d y}{d x} = \frac{\sqrt{1 - y^{2}}}{\sqrt{1 - x^{2}}} \Rightarrow \int \frac{d y}{\sqrt{1 - y^{2}}} = \int \frac{d x}{\sqrt{1 - x^{2}}}$