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CSS Applied Mathematics QUESTION #4145
Question 1
Solve $\frac{dy}{\sqrt{1-y^2}} = \frac{dx}{\sqrt{1-x^2}}$ with $y(0) = \frac{\sqrt{3}}{2}$. What is the solution?
  • $\sin^{-1}(y) = \sin^{-1}(x) + \frac{\pi}{3}$
  • $\sin^{-1}(y) = \sin^{-1}(x) + \frac{\pi}{6}$
  • $y = \sin(\sin^{-1}(x) + \frac{\pi}{3})$
  • Both (A) and (C)✔️
Correct Answer Explanation
Integrating: $\sin^{-1}(y) = \sin^{-1}(x) + C$. At $x=0$, $y=\frac{\sqrt{3}}{2}$: $\frac{\pi}{3} = 0 + C$, so $C = \frac{\pi}{3}$. Thus $\sin^{-1}(y) = \sin^{-1}(x) + \frac{\pi}{3}$, equivalent to $y = \sin(\sin^{-1}(x) + \frac{\pi}{3})$.