If $y = a{e^{mx}} + b{e^{ - mx}}$,then $\frac{d^2y}{dx^2} - m^2y = $

If $y = 100 e^{2x} + 200 e^{-2x}$ and $\frac{d^2 y}{dx^2} = ay$,then $a = $ . . . . . .

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 $\left(\frac{dy}{dx}\right) = \frac{1}{\left(\frac{dx}{dy}\right)}$ 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 $x^m + y^m = k$ where $(m \neq 1)$ and $\frac{d^2y}{dx^2} = \frac{ax^b}{y^c}$ such that $a + b + c = 0$, then the value of $k$ is...

If $a \neq b, x \neq n \pi, n \in Z$ and $y^2 = a^2 \cos^2 x + b^2 \sin^2 x$,then $\frac{d^2 y}{dx^2} + y =$