If $y=e^{\sqrt{x \sqrt{x} \sqrt{x} \ldots}}, x>1$,then $\frac{d^2 y}{d x^2}$ at $x=\log _e 3$ is

If $y = a{x^{n + 1}} + b{x^{ - n}},$ then ${x^2}\frac{{{d^2}y}}{{d{x^2}}}$ is equal to:

If $U_n$ $(n=1,2)$ denotes the $n^{\text{th}}$ derivative of $U(x) = \frac{Lx+M}{x^2-2Bx+C}$ (where $L, M, B, C$ are constants), then the equation $PU_2 + QU_1 + RU = 0$ holds for:

If $f(x) = b e^{ax} + a e^{bx}$,then $f^{\prime \prime}(0)$ is equal to

If $\cos ^{-1}\left(\frac{y}{b}\right)=2 \log \left(\frac{x}{2}\right)$,where $x>0$,then $x^2 \frac{d^2 y}{d x^2}+x \frac{d y}{d x}$ is equal to

If $y = \frac{\log x}{x}$,then the value of $x^2 \frac{d^2 y}{d x^2} + 3x \frac{d y}{d x} + y$ at the point $(\sqrt[3]{e}, \frac{\log(\sqrt[3]{e})}{\sqrt[3]{e}})$ is