If the population grows at the rate $5 \%$ per year,then the time taken for the population to become double is $\quad$ (Given $\log 2 = 0.6912$) (in $years$)

The normal to a curve at $P(x, y)$ meets the $x$-axis at $G$. If the distance of $G$ from the origin is twice the abscissa of $P$,then the curve is

An ice ball melts at a rate proportional to the amount of ice present at that instant. Half of the initial quantity of ice melts in $15 \text{ minutes}$. Let $x_0$ be the initial quantity of ice. If after $30 \text{ minutes}$ the amount of ice left is $k x_0$,then the value of $k$ is:

Number of straight lines which satisfy the differential equation $\frac{dy}{dx} + x \left( \frac{dy}{dx} \right)^2 - y = 0$ is:

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 $2009?$

If the function $y = e^{4x} + 2e^{-x}$ is a solution of the differential equation $\frac{\frac{d^3y}{dx^3} - 13\frac{dy}{dx}}{y} = K$,then the value of $K$ is: