The population of a village increases continuously at a rate proportional to the number of its inhabitants present at any time. If the population of the village was $20,000$ in $1999$ and $25,000$ in the year $2004,$ what will be the population of the village in $2009?$

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

If the general solution of the differential equation $y' = \frac{y}{x} + \phi \left( \frac{x}{y} \right)$,for some function $\phi$,is given by $y \ln |cx| = x$,where $c$ is an arbitrary constant,then $\phi(2)$ is equal to:

If a curve passes through the origin,such that the length of the subnormal is equal to one more than the square of the ordinate,then:

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

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is $1000$ at initial time $t = 0$. The number of bacteria is increased by $20\%$ in $2$ hours. If the population of bacteria is $2000$ after $\frac{k}{\log_{e}\left(\frac{6}{5}\right)}$ hours,then $\left(\frac{k}{\log_{e} 2}\right)^{2}$ is equal to