If ((y)/(x)) = _ e | x | + (α)/(2) is the solution of the...

Given: $\left(xcos\left(\dfrac{y}{x}\right)\right)\dfrac{dy}{dx} = ycos\left(\dfrac{y}{x}\right) + x$

$⇒ \cos \left(\dfrac{y}{x}\right)\dfrac{dy}{dx} = \dfrac{y}{x}\cos \left(\dfrac{y}{x}\right) + 1$

Putting, $y = vx$

$⇒ \dfrac{dy}{dx} = v + x\dfrac{dv}{dx}$

$⇒ \cos v\left(v + x\dfrac{dv}{dx}\right) = vcosv + 1$

$⇒ v + x\dfrac{dv}{dx} = v + \dfrac{1}{\cos v}$

$⇒ x\dfrac{dv}{dx} = \dfrac{1}{\cos v}$

$⇒ \left(\cos v\right)dv = \dfrac{dx}{x}$

$⇒ ∫ \left(\cos v\right)dv = ∫ \dfrac{dx}{x}$

$⇒ \sin v = \log \left|x\right| + c$

$⇒ \sin \left(\dfrac{y}{x}\right) = \log \left|x\right| + c$

Now, $f\left(1\right) = \dfrac{\pi}{3}$

$⇒ \sin \left(\dfrac{\pi}{3}\right) = \log \left|1\right| + c$

$⇒ \dfrac{\sqrt{3}}{2} = c$

$⇒ \sin \left(\dfrac{y}{x}\right) = \log \left|x\right| + \dfrac{\sqrt{3}}{2}$

So, on comparing we get,

$⇒ \alpha = \sqrt{3}$

$⇒ {\alpha}^{2} = 3$