If $y=ae^x+be^{-x}+c$, where $a, b, c$ are parameters, then $y^{\prime \prime \prime}$ is equal to

If $y=e^{ax}(\cos bx+\sin bx)$ satisfies the equation $\frac{d^2y}{dx^2}-K\frac{dy}{dx}+Ly=0$, then $L+bK=$

If $y=5 \cos x-3 \sin x,$ prove that $\frac{d^{2} y}{d x^{2}}+y=0.$

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 $y = (\tan^{-1} x)^2$,then $(x^2 + 1)^2 \frac{d^2 y}{dx^2} + 2x(x^2 + 1) \frac{dy}{dx} = $

If $f(x)=10 \cos x+(13+2 x) \sin x$,then $f^{\prime \prime}(x)+f(x)$ is equal to