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SSC Applied Mathematics QUESTION #1298
Question 1
The method of separation of variables applied to the ODE \(\sqrt{1-y^2}\,dx - \sqrt{1-x^2}\,dy=0\) gives the separated form:
  • \(\dfrac{dx}{\sqrt{1-x^2}} = \dfrac{dy}{\sqrt{1-y^2}}\)✔️
  • \(\sqrt{1-x^2}\,dx = \sqrt{1-y^2}\,dy\)
  • \(\dfrac{dx}{\sqrt{1-y^2}} = \dfrac{dy}{\sqrt{1-x^2}}\)
  • \((1-x^2)dx=(1-y^2)dy\)
Correct Answer Explanation
Rearranging: \(\sqrt{1-y^2}\,dx = \sqrt{1-x^2}\,dy \Rightarrow \dfrac{dx}{\sqrt{1-x^2}}=\dfrac{dy}{\sqrt{1-y^2}}\). Integrating both sides: \(\arcsin x = \arcsin y + C\).