The rate of increase of bacteria in a certain culture is proportional to the number present. If it doubles in $5$ hours,then in $25$ hours,its number would be ......... times the original.

If $y = e^{-mx}$ is a solution of the differential equation $\frac{d^2y}{dx^2} + 4 \frac{dy}{dx} + 3y = 0$, then the values of $m$ are

Let $y = y(x)$ be the solution of the differential equation $x dy - y dx = \sqrt{x^2 - y^2} dx$,$x \geq 1$,with $y(1) = 0$. If the area bounded by the lines $x = 1$,$x = e^{\pi}$,$y = 0$ and the curve $y = y(x)$ is $\alpha e^{2\pi} + \beta$,then the value of $10(\alpha + \beta)$ is equal to ....... .

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 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 $m_{0}$ grams will be reduced to $m_{1}$ grams is (where $k$ is the constant of proportionality):

The solution of $\frac{dy}{dx} = \frac{ax + b}{cy + d}$ represents a parabola if