Question 76

If $$\sin\left(\frac{y}{x}\right) = \log_e|x| + \frac{\alpha}{2}$$ is the solution of the differential equation $$x\cos\left(\frac{y}{x}\right)\frac{dy}{dx} = y\cos\left(\frac{y}{x}\right) + x$$ and $$y(1) = \frac{\pi}{3}$$, then $$\alpha^2$$ is equal to

Substitute $$x = 1, y = \frac{\pi}{3}$$ into the given solution equation:

$$\sin\left(\frac{\pi / 3}{1}\right) = \log_e |1| + \frac{\alpha}{2}$$

$$\frac{\sqrt{3}}{2} = 0 + \frac{\alpha}{2} \implies \alpha = \sqrt{3}$$

$$\alpha^2 = 3$$

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