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

If $y=\tan ^{-1}\left(\frac{\sqrt{1+a^2 x^2}-1}{a x}\right)$,then $\left(1+a^2 x^2\right)y''+2 a^2 x y'$ is equal to

Let $f$ be a twice differentiable function such that $f^{\prime \prime}(x) = -f(x)$,$f^{\prime}(x) = g(x)$,and $h(x) = (f(x))^2 + (g(x))^2$. If $h(5) = 1$,then the value of $h(10)$ is

Find the second order derivative of the function $\sin (\log x)$.

If $x = \sqrt{e^{\sin^{-1} t}}$ and $y = \sqrt{e^{\cos^{-1} t}}$,then $\frac{d^2 y}{dx^2}$ is

If $\sqrt{x+y}-\sqrt{x-y}=c$,then $\frac{d^2 y}{d x^2}=$