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

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

Given $\frac{d^2 y}{d x^2}+\cot x \frac{d y}{d x}+4 y \operatorname{cosec}^2 x=0$. Changing the independent variable $x$ to $z$ by the substitution $z=\log \tan \frac{x}{2}$, the equation is changed to

The population of a town increases at a rate proportional to the population at that time. If the population increases from $40,000$ to $80,000$ in $20$ years,then the population in another $40$ years will be (in $,000$)

The rate of decay of mass of a certain substance at time $t$ is proportional to the mass at that instant. The time during which the original mass of $m_{0} \text{ gm}$ will be reduced to $m_{1} \text{ gm}$ is (where $K$ is the constant of proportionality).

In a certain culture of bacteria,the rate of increase is proportional to the number of bacteria present at that instant. It is found that there are $10,000$ bacteria at the end of $3$ hours and $40,000$ bacteria at the end of $5$ hours. Find the number of bacteria present in the beginning.