If $y = \sin(\log_e x)$,then $x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx}$ is equal to

If $y^2 = P(x)$ is a polynomial of degree $3$,then $2 \left( \frac{d}{dx} \right) \left( y^3 \frac{d^2y}{dx^2} \right)$ equals :

If $x = \int_0^y \frac{1}{\sqrt{1 + 9t^2}} dt$ and $\frac{d^2y}{dx^2} = ay$,then $a$ is equal to

If $y(\theta) = \frac{2 \cos \theta + \cos 2 \theta}{\cos 3 \theta + 4 \cos 2 \theta + 5 \cos \theta + 2}$,then at $\theta = \frac{\pi}{2}$,$y'' + y' + y$ is equal to:

If $y = e^{a \cos^{-1} x}$,$-1 \le x \le 1$,show that $(1-x^{2}) \frac{d^{2} y}{d x^{2}} - x \frac{d y}{d x} - a^{2} y = 0$.

If $x = \log p$ and $y = \frac{1}{p}$,then