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 $y=e^{m \sin ^{-1} x}$, then $(1-x^{2}) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-k y=0$, where $k$ is equal to

If $y^{2}=a x^{2}+b x+c$,where $a, b, c$ are constants,then $y^{3} \frac{d^{2} y}{d x^{2}}$ is equal to

Let $f$ and $g$ be twice differentiable functions such that $f(x) \cdot g(x) = 1$ for all $x \in R$ and $f'$ and $g'$ are never zero. Then $\frac{f''(x)}{f(x)} + \frac{g''(x)}{g(x)}$ equals:

If $y = a \cos(\log x) + b \sin(\log x)$ where $a, b$ are parameters,then ${x^2}y'' + xy' = $

If $f(x)$ and $g(x)$ are twice differentiable functions on $(0,3)$ satisfying $f^{\prime \prime}(x)=g^{\prime \prime}(x)$, $f^{\prime}(1)=4$, $g^{\prime}(1)=6$, $f(2)=3$, and $g(2)=9$, then $f(1)-g(1)$ is