JEE Mains · Maths · STD 12 - 9. differential equations
Let \(y=y(x)\) be the solution of the differential equation \(x\sqrt{1-x^2}\,dy + \left(y\sqrt{1-x^2} - x\cos^{-1}x\right)dx = 0\), \(x \in (0, 1)\), \(\displaystyle\lim_{x\to 1^-} y(x) = 1\). Then \(y\left(\dfrac{1}{2}\right)\) equals:
- A \(3 - \dfrac{\pi}{\sqrt{3}}\)
- B \(4 - \sqrt{3}\pi\)
- C \(4 - \dfrac{2\pi}{\sqrt{3}}\)
- D \(3 - \dfrac{\pi}{2\sqrt{3}}\)
Answer & Solution
Correct Answer
(A) \(3 - \dfrac{\pi}{\sqrt{3}}\)
Step-by-step Solution
Detailed explanation
The given differential equation is: \(x\sqrt{1-x^2}\,dy + \left(y\sqrt{1-x^2} - x\cos^{-1}x\right)dx = 0\) Rearranging the terms, we get: \(x\sqrt{1-x^2}\,dy + y\sqrt{1-x^2}\,dx = x\cos^{-1}x\,dx\) Dividing the entire equation by \(\sqrt{1-x^2}\):…
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