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CSS Applied Mathematics QUESTION #1299
Question 1
The solution of \(\sqrt{1-y^2}\,dx-\sqrt{1-x^2}\,dy=0\) with \(y(0)=\dfrac{\sqrt{3}}{2}\) is:
  • \(\arcsin x - \arcsin y = \dfrac{\pi}{3}\)
  • \(\arcsin x = \arcsin y - \dfrac{\pi}{3}\)✔️
  • \(x + y = \sqrt{3}\)
  • \(\arcsin x + \arcsin y = \pi\)
Correct Answer Explanation
Integrating: \(\arcsin x = \arcsin y + C\). At \(x=0\), \(y=\dfrac{\sqrt{3}}{2}\): \(0=\arcsin\dfrac{\sqrt{3}}{2}+C=\dfrac{\pi}{3}+C\Rightarrow C=-\dfrac{\pi}{3}\). So \(\arcsin x=\arcsin y-\dfrac{\pi}{3}\).