If $y=3 \cos (\log x)+4 \sin (\log x)$,then $x^2 y_2+x y_1=$

If $f(x) = \sin x \cdot \sin 2x \cdot \sin 3x$ and $f''(x) = a(\sin bx) + c(\sin dx) + e(\sin kx)$,then the value of $(a+c+e) - (b+d+k)$ equals

If $y = a{e^{mx}} + b{e^{ - mx}}$,then $\frac{d^2y}{dx^2} - m^2y = $

If $y_k$ is the $k$-th derivative of $y$ with respect to $x$,and $y = \cos(\sin x)$,then $y_1 \sin x + y_2 \cos x$ is equal to

Suppose the function $f(x) - f(2x)$ has the derivative $5$ at $x = 1$ and derivative $7$ at $x = 2$. The derivative of the function $f(x) - f(4x)$ at $x = 1$ has the value equal to:

If $y=2 x^{n+1}+\frac{3}{x^{n}}$,then $x^{2} \frac{d^{2} y}{d x^{2}}$ is