If a curve $ y = y(x) $ passes through the point $ \left(1, \frac{\pi}{2}\right) $ and satisfies the differential equation
$$
(7x^4 \cot y - e^x \csc y) \frac{dx}{dy} = x^5, \quad x \geq 1, \text{ then at } x = 2, \text{ the value of } \cos y \text{ is:}
$$
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In solving differential equations, use separation of variables and integration to solve for \( y \), then use the given point to find constants of integration.
The differential equation provided is: \[ (7x^4 \cot y - e^x \csc y) \frac{dx}{dy} = x^5 \] This is rearranged to: \[ \frac{dx}{dy} = \frac{x^5}{7x^4 \cot y - e^x \csc y} \] For variable separation, we invert to \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{7x^4 \cot y - e^x \csc y}{x^5} \] Integration is performed after evaluating at \( x = 1 \) and \( x = 2 \). We focus on \( \cos y \) at \( x = 2 \). The calculated value of \( \cos y \) at \( x = 2 \) is \( \frac{e^2}{128} - 1 \). Therefore, the correct answer is \( \frac{e^2}{128} - 1 \).
