$y = a \cos x + (b + 2x) \sin x \Rightarrow y^{\prime \prime} + y = $

If $p(x)$ is a polynomial of degree $3$ which satisfies $p^{\prime \prime}(1)=0$ and $p^{\prime \prime \prime}(1)=6$,then $p^{\prime \prime}(0)$ is equal to

If $y = x^2 e^{mx}$,where $m$ is a constant,then $\frac{d^3y}{dx^3} = $

If $f(x) = a\sin (\log x)$,then ${x^2}f''(x) + xf'(x) = . . . $

There exists a function $f(x)$ satisfying $f(0) = 1$,$f'(0) = -1$,and $f(x) > 0$ for all $x$. Which of the following must be true for $f''(x)$?

If $y = (\sin^{-1} x)^2 + (\cos^{-1} x)^2$,then $(1 - x^2) y_2 - x y_1 =$