$\sin\!\left(\frac{y}{x}\right)\frac{dy}{dx} = \frac{y}{x}\sin\!\left(\frac{y}{x}\right)-1$

put $y=tx \Rightarrow \frac{dy}{dx} = t+x\frac{dt}{dx}$

$\sin t\left(t+x\frac{dt}{dx}\right) = t\sin t-1$

$x\sin t\frac{dt}{dx}+1=0$

$\sin tdt+\frac{dx}{x}=0$

$\Rightarrow -\cos t+\ln x=C$

$\Rightarrow -\cos\!\left(\frac{y}{x}\right)+\ln x=C$

$y(1)=\frac{\pi}{2} \Rightarrow C=0$

$\Rightarrow \cos\!\left(\frac{y}{x}\right)=\ln x$

$\csc\!\left( \frac{y(e^{12})}{e^{12}} \right) = 12 \Rightarrow \alpha=12$

$x^2+y^2-2px+2py+14=0$

$r=\sqrt{p^2+p^2-14}$

$\Rightarrow r\le 6 \Rightarrow r^2\le 36$

$2p^2-14\le 36$

$p^2\le 25$

$p\in [-5,5]$

number of integral value of $p=11$