Let $S$ be the set of real numbers $p$ such that there is no non-zero continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $\int_0^x f(t) dt = p f(x)$ for all $x \in \mathbb{R}$. Then,$S$ is

The bacteria increase at a rate proportional to the number of bacteria present. If the original number $N$ doubles in $8$ hours,then the number of bacteria in $24$ hours will be (in $N$)

The solution of the differential equation $\frac{dy}{dx} + \frac{x}{y} \cdot \frac{x^2+y^2-1}{2(x^2+y^2)+1} = 0$ is

The differential equation $\frac{dy}{dx} = \frac{\sqrt{1-y^2}}{y}$ determines a family of circles with

The solution of the differential equation $y \frac{dy}{dx} + x = k$ represents . . . . . . .

Let $I$ be the purchase value of an equipment and $V(t)$ be the value after it has been used for $t$ years. The value $V(t)$ depreciates at a rate given by the differential equation $\frac{dV(t)}{dt} = -k(T - t)$,where $k > 0$ is a constant and $T$ is the total life in years of the equipment. Then the scrap value $V(T)$ of the equipment is: