JEE Mains · Maths · STD 12 - 9. differential equations
Let \(y=y(x)\) be the solution of the differential equation \(\left(1-x^2\right) d y=\left[x y+\left(x^3+2\right) \sqrt{3\left(1-x^2\right)}\right] d x\) \(-1 < x < 1, y(0)=0\). If \(y\left(\frac{1}{2}\right)=\frac{\mathrm{m}}{\mathrm{n}}, \mathrm{m}\) and \(\mathrm{n}\) are coprime numbers, then \(\mathrm{m}+\mathrm{n}\) is equal to ...........
- A \(91\)
- B \(92\)
- C \(97\)
- D \(77\)
Answer & Solution
Correct Answer
(C) \(97\)
Step-by-step Solution
Detailed explanation
\( \frac{d y}{d x}-\frac{x y}{1-x^2}=\frac{\left(x^3+2\right) \sqrt{3\left(1-x^2\right)}}{1-x^2} \) \( I F=e^{-\int \frac{x}{1-x^2} d x}=e^{+\frac{1}{2} \ln \left(1-x^2\right)}=\sqrt{1-x^2} \) \( y \sqrt{1-x^2}=\sqrt{3} \int\left(x^3+2\right) d x \)…
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