JEE Mains · Maths · STD 12 - 9. differential equations
Let \(y=y(x)\) be the solution of the differential equation \( \frac{d y}{d x}=\frac{(\tan x)+y}{\sin x(\sec x-\sin x \tan x)}\), \(x \in\left(0, \frac{\pi}{2}\right)\) satisfying the condition \(y\left(\frac{\pi}{4}\right)=2\). Then, \(y\left(\frac{\pi}{3}\right)\) is
- A \(\sqrt{3}\left(2+\log _{\mathrm{e}} \sqrt{3}\right)\)
- B \(\frac{\sqrt{3}}{2}\left(2+\log _e 3\right)\)
- C \(\sqrt{3}\left(1+2 \log _e 3\right)\)
- D \(\sqrt{3}\left(2+\log _e 3\right)\)
Answer & Solution
Correct Answer
(A) \(\sqrt{3}\left(2+\log _{\mathrm{e}} \sqrt{3}\right)\)
Step-by-step Solution
Detailed explanation
\(\frac{d y}{d x}=\frac{\sin x+y \cos x}{\sin x \cdot \cos x\left(\frac{1}{\cos x}-\sin x \cdot \frac{\sin x}{\cos x}\right)}\) \(=\frac{\sin x+y \cos x}{\sin x\left(1-\sin ^2 x\right)}\) \(\frac{d y}{d x}=\sec ^2 x+y \cdot 2(\operatorname{cosec} 2 x)\)…
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