If $x = \int\limits_0^y {\frac{{dt}}{{\sqrt {1 + {t^2}} }}} $,then $\frac{{{d^2}y}}{{d{x^2}}}$ 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

$A$ function $f$ satisfies the relation $f(x) = f''(x) + f'''(x) + \dots \infty$,where $f(x)$ is an infinitely differentiable function. If $f(1) = 5$,then the value of $f'(1) + f''(1)$ 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

$f(x)$ is a twice differentiable function such that $f^{\prime \prime}(x)=-f(x)$ and $f^{\prime}(x)=g(x)$. If $h(x)=(f(x))^2+(g(x))^2$ and $h(1)=2$, then $h(2)=$

If $\cos ^{-1}\left(\frac{y}{2}\right)=\log _{e}\left(\frac{x}{5}\right)^{5}$ and $|y| < 2$,then which of the following is true?