If $y = a^x \cdot b^{2x - 1}$,then $\frac{d^2y}{dx^2}$ is

The function $y=e^{kx}$ satisfies $(\frac{d^2y}{dx^2}+\frac{dy}{dx})(\frac{dy}{dx}-y)=y\frac{dy}{dx}$. It is valid for

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 $y = \operatorname{Sin}^{-1}\left(\frac{2x}{1+x^2}\right)$ and $\left(\frac{d^2y}{dx^2}\right)_{x=2} = k$,then $25k =$

$y = a \cos x + (b + 2x) \sin x \Rightarrow y^{\prime \prime} + y = $

Find the value of $\frac{d^2x}{dy^2}$.