Let y=y(x) be the solution of the differential equation x sin (y/x) d y=(y sin (y/x)-x) d x, y(1)=(π)/2 and let α= cos ((y(e^12))/(e^12)). Then the…

Let $\displaystyle y=y(x)$ be the solution of the differential equation $\displaystyle x \sin \left(\frac{y}{x}\right) d y=\left(y \sin \left(\frac{y}{x}\right)-x\right) d x, y(1)=\frac{\pi}{2}$ and let $\displaystyle \alpha=\cos \left(\frac{y\left(e^{12}\right)}{e^{12}}\right)$. Then the number of integral value of $\displaystyle p$, for which the equation $\displaystyle x^2+y^2-2 p x+2 p y+\alpha+2=0$ represents a circle of radius $\displaystyle r \leq 6$, is $\displaystyle \_\_\_\_$.