If \(y = y(x)\) is the solution of the differential equation \(\displaystyle\frac{dy}{dx} = \sqrt{\frac{1 - y^2}{1 - x^2}}\), \(-1 \lt x \lt 1\), satisfying \(y(0) = \displaystyle\frac{1}{2}\), then \(y\left(\displaystyle\frac{1}{2}\right)\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{\sqrt{3}}{2}\)
The equation is variables separable:
\( \displaystyle \frac{dy}{\sqrt{1 - y^2}} = \frac{dx}{\sqrt{1 - x^2}} \)
Integrating both sides:
\(\sin^{-1} y = \sin^{-1} x + C\)
Putting x = 0, y = 1/2:
\( \displaystyle C = \sin^{-1}\frac{1}{2} = \frac{\pi}{6} \)
At \(x = \frac{1}{2}\):
\( \displaystyle \sin^{-1} y = \frac{\pi}{6} + \frac{\pi}{6} = \frac{\pi}{3} \)
\( \displaystyle y = \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2} \)