Let $y = \log(x + \sqrt{x^2+1})$, and $a\frac{d^2y}{dx^2} + b\frac{dy}{dx} = c$. Then identify the correct statements about the values of $a$, $b$ and $c$ :

(A) $a = 1 + x^2$

(B) $b = 0$

(C) $c = 0$

(D) $b = x$

(E) $c = 2$

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Match List-I with List-II

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If $\sqrt{1-x^2} + \sqrt{1-y^2} = a(x-y)$, then $\frac{dy}{dx} =$

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 $P(x)-xQ(x)-4\alpha=0$

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If $f(x) = 2x$ and $g(x) = \frac{x^2}{2} + 1$, then which of the following can be a discontinuous function ?

Match List-I with List-II

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If $y=e^{2\log_e t}$ and $x=\log_e e^{t^2}$, then the value of $\frac{d^2y}{dx^2}$ is