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 $x^{2} y^{5}=(x+y)^{7}$,then $\frac{d^{2} y}{d x^{2}}$ is equal to

If ${x^p}{y^q} = {(x + y)^{p + q}}$,then $\frac{{{d^2}y}}{{d{x^2}}} = $

Variables $x$ and $y$ are related by the equation $x = \int\limits_0^y \frac{dt}{\sqrt{1 + t^2}}$. The value of $\frac{d^2y}{dx^2}$ is equal to

If $y^2 = P(x)$ is a polynomial of degree $3$,then $2 \left( \frac{d}{dx} \right) \left( y^3 \frac{d^2y}{dx^2} \right)$ equals :

If $f(x)$ and $g(x)$ are twice differentiable functions satisfying $f''(x) = g''(x)$, $f'(1) = 2g'(1) = 4$ and $f(2) = 3g(2) = 9$, then the value of $f(x) - g(x)$ at $x = 4$ is: