If $y = {\left[ {x + \sqrt {{x^2} - 1} } \right]^{15}} + {\left[ {x - \sqrt {{x^2} - 1} } \right]^{15}}$,then $\left( {{x^2} - 1} \right)\frac{{{d^2}y}}{{d{x^2}}} + x\frac{{dy}}{{dx}}$ is equal to

If $y = \cos^{-1} x$,find $\frac{d^{2} y}{d x^{2}}$ in terms of $y$ alone.

If $f: R \rightarrow R$ is an even function which is twice differentiable on $R$ and $f^{\prime \prime}(\pi)=1$, then $f^{\prime \prime}(-\pi)$ is equal to

If the three functions $f(x)$,$g(x)$,and $h(x)$ are such that $h(x) = f(x) \cdot g(x)$ and $f^{\prime}(x) \cdot g^{\prime}(x) = c$,where $c$ is a constant,then $\frac{f^{\prime \prime}(x)}{f(x)} + \frac{g^{\prime \prime}(x)}{g(x)} + \frac{2c}{f(x) \cdot g(x)}$ is equal to:

If $y = (\cos^{-1} x)^2$,then $(1-x^2) y_2 + p x y_1 + q = 0$. Find the value of $p+q$.

If $y = a{x^{n + 1}} + b{x^{ - n}}$,then ${x^2}\frac{{{d^2}y}}{{d{x^2}}} = $