If $y=\tan ^{-1}\left(\frac{\sqrt{1+a^2 x^2}-1}{a x}\right)$, then $\left(1+a^2 x^2\right)y''+2 a^2 x y'$ is equal to

If $f(x)=10 \cos x+(13+2 x) \sin x$,then $f^{\prime \prime}(x)+f(x)$ is equal to

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 $f : S \rightarrow S$ where $S =(0, \infty)$ be a twice differentiable function such that $f(x+1) = xf(x)$. If $g : S \rightarrow R$ is defined as $g(x) = \log_{e} f(x)$,then the value of $|g''(5) - g''(1)|$ is equal to:

If $y = \sin(\sin x)$ and $y'' + f(x) \cdot y' + g(x) \cdot y = 0$,then $f(x) \cdot g(x) =$

If $x+1=e^{-y}$,then $\frac{d^2 y}{d x^2} = $ . . . . . .