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

If $x=\frac{1-\sqrt{y}}{1+\sqrt{y}}$, then $(x+1) \frac{d^2 y}{d x^2}+\left(\frac{3 \sqrt{y}+1}{\sqrt{y}}\right) \frac{d y}{d x}$ equals

If $U_n$ $(n=1,2)$ denotes the $n^{\text{th}}$ derivative of $U(x) = \frac{Lx+M}{x^2-2Bx+C}$ (where $L, M, B, C$ are constants), then the equation $PU_2 + QU_1 + RU = 0$ holds for:

If $y=e^{\sin ^{-1} x}$,then $\left(1-x^2\right) y_2-x y_1=$

If $\left(\frac{dy}{dx}\right) = \frac{1}{\left(\frac{dx}{dy}\right)}$ and $\frac{d^2x}{dy^2}\left(\frac{dy}{dx}\right)^3 + \frac{d^2y}{dx^2} = k$, then $e^{k f(x)} - k f(x) =$

If $y = 1 + x + x^2 + x^3 + . . . + \infty$ and $|x| < 1$,then $y^{\prime \prime} =$