JEE Mains · Maths · STD 12 - 9. differential equations
Let \(v\) be the solution of the differential equation \(\left(1-x^{2}\right) d y=\left(xy+\left(x^{3}+2\right) \sqrt{1-x^{2}}\right) d x,-1 < x < 1\) and \(y(0)=0\) if \(\int\limits_{-\frac{1}{2}}^{\frac{1}{2}} \sqrt{1-x^{2}} y(x) d x=k\) then \(k^{-1}\) is equal to:
- A \(320\)
- B \(321\)
- C \(322\)
- D \(323\)
Answer & Solution
Correct Answer
(C) \(322\)
Step-by-step Solution
Detailed explanation
\(\left(1-x^{2}\right) \frac{d y}{d x}=x y+\left(x^{3}+2\right) \sqrt{1-x^{2}}\) \(\Rightarrow \frac{d y}{d x}+\left(\frac{-x}{1-x^{2}}\right) y=\frac{x^{3}+2}{\sqrt{1-x^{2}}}\) \(I F=e^{\int \frac{-x}{1-x^{2}} d x}=\sqrt{1-x^{2}}\)…
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