If $y=a \cos (\log x)+b \sin (\log x)$, where $a, b$ are parameters, then $x^2 y^{\prime \prime}+x y^{\prime}$ is equal to

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

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

Let $g(x) = \log(f(x))$ where $f(x)$ is a twice differentiable positive function on $(0, \infty)$ such that $f(x+1) = x f(x)$. Then,for $N = 1, 2, 3, \ldots$,$g^{\prime \prime}\left(N+\frac{1}{2}\right) - g^{\prime \prime}\left(\frac{1}{2}\right) = $

If $f(x)$ is a polynomial such that $f(x) = [f'(x)]^2$ and $f(2) = 0$, then $f(-2) = ...$

If $y = x + e^x$,then at $x = 1$,$\frac{d^2x}{dy^2}$ is equal to