For the function $y(x)=\sqrt{x+1}-\sqrt{x-1}$, which of the following are true?

(A) $\left(\frac{dy}{dx}\right)_{at\,x=2}=\frac{1-\sqrt{3}}{2\sqrt{3}}$

(B) $(x^2-1)\frac{d^2y}{dx^2}-\frac{dy}{dx}+4y=0$

(C) $(x^2-1)\left(\frac{dy}{dx}\right)^2-4y^2=0$

(D) If $P(x)\frac{d^2y}{dx^2}+Q(x)\frac{dy}{dx}+\alpha y=0$, then $P(x)-xQ(x)-4\alpha=0$

Choose the correct answer from the options given below:

If $y^2 = ax + b, a,b \in \mathbb{R}$ where $\mathbb{R}$ is the set of real numbers, then $y\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2$ is equal to:

If $x^2+y^2 = t+\dfrac{1}{t}$ and $x^4+y^4=t^2+\dfrac{1}{t^2}$, then $\dfrac{dy}{dx}$ is equal to

If $y = e^{acos^{-1}x}, -1 < x < 1$, then $(1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx}$ is equal to

If $y = 3e^{2x} + 2e^{3x}$, then $\frac{d^2y}{dx^2} + 6y$ is equal to

Let $y = \log_e \left(\frac{a + b \sin x}{a - b \sin x}\right)$, then value of $\frac{dy}{dx}$ is :

If $f(x) = \begin{cases} \frac{\tan(\frac{\pi}{4} - x)}{\cot 2x} & , x ≠ \frac{\pi}{4} \\ 2K + 1 & , x = \frac{\pi}{4} \end{cases}$ is continuous at $x = \frac{\pi}{4}$, then the value of K is equal to