JEE Mains · Maths · STD 12 - 9. differential equations
If a curve \(y=y(x)\) passes through the point \(\left(1, \frac{\pi}{2}\right)\) and satisfies the differential equation \(\left(7 x^4 \cot y-e^x \operatorname{cosec} y\right) \frac{d x}{d y}=x^5, x \geq 1\), then at \(x=2\), the value of cosy is:
- A \(\frac{2 \mathrm{e}^2-\mathrm{e}}{64}\)
- B \(\frac{2 \mathrm{e}^2+\mathrm{e}}{64}\)
- C \(\frac{2 \mathrm{e}^2-\mathrm{e}}{128}\)
- D \(\frac{2 \mathrm{e}^2+\mathrm{e}}{128}\)
Answer & Solution
Correct Answer
(C) \(\frac{2 \mathrm{e}^2-\mathrm{e}}{128}\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \frac{d y}{d x}=\frac{7 \cot y}{x}-\frac{e^x \operatorname{cosec} y}{x^5} \\ & \frac{d y}{d x}=\frac{7 \cot y}{\sin y \cdot x}-\frac{e^x}{\sin y x^5} \\ & \sin y \frac{d y}{d x}-\cos y \cdot \frac{7}{x}=\frac{-e^x}{x^5} \\ & \text { let }-\cos y=t \\ & \sin y…
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