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The solution curve y = y(x) of the differential equation [x / sqrt(x² - y²) + e^y] * (dy/dx) = x + [x / sqrt(x² - y²) + e^x] * y passes through (1, 0) and (2*alpha, alpha), where alpha > 0. Then alpha equals:

1. (1/2) * exp(pi/6 + sqrt(e) - 1)
2. (1/2) * exp(pi/3 + sqrt(e) - 1)
3. exp(pi/6 + sqrt(e) + 1)
4. 2 * exp(pi/3 + sqrt(e) - 1)

Correct answer: (1/2) * exp(pi/3 + sqrt(e) - 1)

Solution

This JEE Advanced 2022 problem involves recognizing exact differentials. After careful rearrangement, the equation integrates to e^y/x + arcsin(y/x) = sqrt(e^x)/x + constant (schematically). Applying the boundary conditions (1,0) and (2*alpha, alpha) yields alpha = (1/2)*exp(pi/3 + sqrt(e) - 1).

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