If $f(x) = 10 \cos x + (13 + 2x) \sin x$,then $f''(x) + f(x)$ is equal to

$\frac{d^n}{dx^n}(e^{2x} + e^{-2x}) = $

Let $f$ be a polynomial function such that $f(3x) = f'(x) \cdot f''(x)$ for all $x \in \mathbb{R}$. Then:

Let $f(x)$ and $g(x)$ be two functions having finite non-zero $3^{rd}$ order derivatives $f'''(x)$ and $g'''(x)$ for all $x \in R$. If $f(x)g(x) = 1$ for all $x \in R$,then $\frac{f'''}{f'} - \frac{g'''}{g'}$ is equal to

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

If $(a+b x) e^{\frac{y}{x}}=x$,then $\frac{d^2 y}{d x^2}=$